#6372113

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2134 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)(8,9)(11,12)(13,14)(15,20)(16,28)(17,22)(18,32)(19,23)(21,31)(24,37)(25,34)(26,35)(27,41)(29,42)(30,36)(33,39)(38,43)(40,44);(1,3,4,2)(6,9)(10,12)(13,15)(14,20)(16,24)(17,26)(18,27)(19,25)(21,33)(22,35)(23,34)(28,37)(29,40)(30,38)(31,39)(32,41)(36,43)(42,44);(13,16,18,31,19,29,17,30)(14,21,22,28,23,36,32,42)(15,24,27,39,25,40,26,38)(20,33,35,37,34,43,41,44);(5,7,8,11)(6,10,9,12)(15,25)(16,29)(20,34)(21,36)(26,27)(28,42)(30,31)(35,41);(13,17,19,18)(14,22,23,32)(15,26,25,27)(16,30,29,31)(20,35,34,41)(21,28,36,42)(24,38,40,39)(33,37,43,44)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=di,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279309);