#6403905

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1848 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6311101);