#7561491

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,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,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, 7468687);