#496709

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403905);