#6389009

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1966 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)(7,10)(8,11)(9,12)(13,14,18,22)(15,23,24,36)(16,20,27,34)(17,29,31,21)(19,30,32,26)(25,37,38,44)(28,41,40,35)(33,42,43,39);(1,3,4,2)(5,7,10,6)(8,11,9,12)(15,24)(19,32)(23,36)(25,38)(26,30)(33,43)(37,44)(39,42);(13,15,18,24)(14,19,22,32)(16,25,27,38)(17,30,31,26)(20,33,34,43)(21,36,29,23)(28,42,40,39)(35,44,41,37);(5,8,10,9)(6,11,7,12)(13,16,18,27)(14,20,22,34)(15,25,24,38)(17,28,31,40)(19,33,32,43)(21,35,29,41)(23,37,36,44)(26,39,30,42);(5,9,10,8)(6,11,7,12)(16,27)(20,34)(25,38)(28,40)(33,43)(35,41)(37,44)(39,42)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296205);