#7561600

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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

2203 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)(13,14,19,24)(15,25,16,21)(17,22,32,41)(18,33,20,23)(26,39,28,35)(27,36,29,37)(30,44,43,38)(31,42,34,40);(2,3)(6,10)(11,12)(13,15,19,16)(14,21,24,25)(17,30,32,43)(18,27,20,29)(22,38,41,44)(23,36,33,37)(26,34,28,31)(35,40,39,42);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,19,15)(14,21,24,25)(17,26,32,28)(18,27,20,29)(22,39,41,35)(23,37,33,36)(30,34,43,31)(38,42,44,40);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17,19,32)(14,22,24,41)(15,26,16,28)(18,31,20,34)(21,35,25,39)(23,40,33,42)(27,43,29,30)(36,38,37,44);(6,10)(11,12)(13,18,19,20)(14,23,24,33)(15,27,16,29)(17,31,32,34)(21,36,25,37)(22,40,41,42)(26,43,28,30)(35,38,39,44)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7468796);