#513074

order	512
nilpotency class	5
isoclinism	isoclinic class of #512941
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

339 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(11,21)(12,22)(13,26)(14,29)(16,31)(18,32)(19,39)(20,38)(23,48)(24,51)(25,44)(27,53)(28,47)(30,52)(33,54)(34,63)(35,64)(36,58)(37,65)(40,66)(41,42)(43,49)(45,46)(50,69)(55,56)(57,62)(59,60)(61,72)(67,68)(70,71);(2,5)(6,8)(9,10,15,17)(11,16,25,33)(12,18,28,36)(13,19,30,37)(14,20,29,38)(21,31,44,54)(22,32,47,58)(23,34,50,61)(24,35,49,62)(26,39,52,65)(27,40,45,59)(41,55,67,70)(42,56,68,71)(43,57,51,64)(46,60,53,66)(48,63,69,72);(9,11,22,42,13,23,45,51,29,49,53,69,52,67,28,44,15,25,47,68,30,50,27,43,14,24,46,48,26,41,12,21)(10,16,32,56,19,34,59,64,38,62,66,72,65,70,36,54,17,33,58,71,37,61,40,57,20,35,60,63,39,55,18,31);(1,3,4,7)(2,6,5,8)(10,17)(16,33)(18,36)(19,37)(20,38)(31,54)(32,58)(34,61)(35,62)(39,65)(40,59)(55,70)(56,71)(57,64)(60,66)(63,72)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232017, #72321786, #72322373, …
maximal normal subgroup of	#72232017, #72321786, #72322373, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420270);