#149418

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,g^b=gi

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, 56614);