#7557572

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

1754 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)(5,6,9,11)(7,10,12,8)(13,14,17,21)(15,23,26,19)(16,20,28,33)(18,22,30,35)(24,37,39,31)(25,32,27,36)(29,34,41,43)(38,42,40,44);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,31)(21,23)(22,36)(27,30)(28,39)(29,38)(32,35)(33,37)(34,44)(40,41)(42,43);(6,11)(7,12)(13,16,17,28)(14,20,21,33)(15,24,26,39)(18,29,30,41)(19,31,23,37)(22,34,35,43)(25,38,27,40)(32,42,36,44);(5,8,9,10)(6,12,11,7)(14,21)(15,25)(18,30)(19,32)(20,33)(23,36)(24,38)(26,27)(29,41)(31,42)(37,44)(39,40);(1,3,4,2)(14,21)(15,26)(16,28)(22,35)(25,27)(29,41)(31,37)(42,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=i,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, 7464768);