#493317

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,h^2=i,b^a=be,c^a=ci,d^a=dei,c^b=ce,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400513);