#7185543

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#30549	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1839 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)(6,10)(8,11)(13,14,19,23)(15,20,18,32)(16,26,28,39)(17,22,30,36)(21,24,35,29)(25,33,31,42)(27,38,41,44)(34,37,43,40);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(13,15,19,18)(14,20,23,32)(16,24,28,29)(17,25,30,31)(21,26,35,39)(22,33,36,42)(27,37,41,40)(34,38,43,44);(5,7)(6,11)(8,10)(9,12)(13,16,15,24,19,28,18,29)(14,21,32,39,23,35,20,26)(17,27,25,37,30,41,31,40)(22,34,42,44,36,43,33,38);(2,4)(6,10)(7,12)(13,17,19,30)(14,22,23,36)(15,25,18,31)(16,27,28,41)(20,33,32,42)(21,34,35,43)(24,37,29,40)(26,38,39,44);(5,8,9,11)(6,7,10,12)(16,28)(17,30)(21,35)(22,36)(24,29)(25,31)(26,39)(33,42)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092739);