#7614380

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1867 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=hi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

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, 7521576);