#7561466

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

2200 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)(13,14,19,23)(15,20,26,36)(16,21,29,40)(17,30,18,22)(24,33,42,43)(25,35,31,34)(27,37,41,44)(28,39,32,38);(2,3)(6,10)(11,12)(13,15)(14,20)(16,27)(17,25)(18,31)(19,26)(21,37)(22,34)(23,36)(24,28)(29,41)(30,35)(32,42)(33,39)(38,43)(40,44);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15)(14,20)(16,24)(17,31)(18,25)(19,26)(21,33)(22,35)(23,36)(27,28)(29,42)(30,34)(32,41)(37,39)(38,44)(40,43);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,16,19,29)(14,21,23,40)(15,24,26,42)(17,28,18,32)(20,33,36,43)(22,38,30,39)(25,41,31,27)(34,37,35,44);(6,10)(11,12)(13,17,19,18)(14,22,23,30)(15,25,26,31)(16,28,29,32)(20,34,36,35)(21,38,40,39)(24,41,42,27)(33,37,43,44)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fgi,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, 7468662);