#7503707

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1970 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410903);