#7561464

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

2236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^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=efi,c^b=cg,d^b=dh,h^b=hi,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, 7468660);