#6467649

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#28349	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,20,18,24,17,23)(15,25,32,37,28,21,26,34)(16,22,33,35,31,39,29,40)(27,38,42,44,30,36,41,43);(2,4)(14,20)(15,26)(16,29)(17,19)(23,24)(27,30)(28,32)(31,33)(34,37)(35,40)(36,44)(38,43);(6,9)(10,12)(13,15,17,26,18,28,19,32)(14,21,23,37,24,25,20,34)(16,30,29,42,31,27,33,41)(22,38,40,43,39,36,35,44);(13,16,17,29,18,31,19,33)(14,22,23,40,24,39,20,35)(15,27,26,41,28,30,32,42)(21,36,37,44,25,38,34,43);(5,7,8,11)(6,10,9,12)(16,31)(22,39)(27,30)(29,33)(35,40)(36,38)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=h,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374845);