#6389874

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
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×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2188 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)(15,20)(16,27)(17,22)(18,32)(19,23)(21,30)(24,38)(25,34)(26,36)(28,41)(29,37)(31,35)(33,40)(39,44)(42,43);(1,3,4,2)(5,7,10,6)(8,11,9,12)(13,15,19,26)(14,20,23,36)(16,24,28,39)(17,31,18,25)(21,33,37,43)(22,35,32,34)(27,38,41,44)(29,42,30,40);(13,16,18,30,19,28,17,29)(14,21,22,27,23,37,32,41)(15,24,25,40,26,39,31,42)(20,33,35,38,36,43,34,44);(5,8,10,9)(6,11,7,12)(16,28)(21,37)(24,39)(27,41)(29,30)(33,43)(38,44)(40,42);(5,9,10,8)(6,11,7,12)(13,17,19,18)(14,22,23,32)(15,25,26,31)(16,29,28,30)(20,34,36,35)(21,27,37,41)(24,40,39,42)(33,44,43,38)

Polycyclic presentation:

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

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, 6297070);