#6372105

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

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,c^b=ci,d^b=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, 6279301);