#7054369

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2110 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961565);