#493270

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×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

822 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)(13,14,16,21,17,22,26,34,18,20,27,32,28,33,38,42)(15,19,24,30,25,31,37,41,23,29,35,39,36,40,43,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(15,23)(19,29)(24,35)(25,36)(30,39)(31,40)(37,43)(41,44);(7,12)(9,11)(13,15)(14,19)(16,24)(17,25)(18,23)(20,29)(21,30)(22,31)(26,37)(27,35)(28,36)(32,39)(33,40)(34,41)(38,43)(42,44);(6,10)(9,11)(14,20)(15,23)(21,32)(22,33)(24,35)(25,36)(34,42)(37,43)

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,h^2=i,b^a=be,d^a=dei,c^b=cei,d^c=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, 400466);