#491440

order	512
nilpotency class	4
isoclinism	isoclinic class of #491439
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

560 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 182 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=f,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398636);