#7527733

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
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1965 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)(5,6,8,10)(7,9,11,12)(13,14)(15,21)(16,22)(17,31)(18,24)(19,23)(20,27)(25,37)(26,35)(28,44)(29,41)(30,40)(32,36)(33,39)(34,43)(38,42);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,28)(17,26)(18,27)(19,32)(21,24)(22,38)(23,35)(25,29)(30,42)(31,36)(33,43)(34,41)(37,39)(40,44);(6,10)(7,11)(13,15)(14,21)(16,25)(17,32)(18,27)(19,26)(20,24)(22,37)(23,35)(28,29)(30,43)(31,36)(33,42)(34,40)(38,39)(41,44);(6,10)(9,12)(13,16,18,30)(14,22,24,40)(15,25,27,43)(17,29,19,33)(20,34,21,37)(23,39,31,41)(26,42,32,28)(35,38,36,44);(1,3,4,2)(13,17,18,19)(14,23,24,31)(15,26,27,32)(16,29,30,33)(20,35,21,36)(22,39,40,41)(25,42,43,28)(34,38,37,44)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=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, 7434929);