#108400

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

654 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,f^2=i,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^c=fh

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, 15596);