#356915

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=fg,e^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=ei,g^b=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264111);