#7561743

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2178 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7468939);