#105300

order	512
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₂²×C₄×C₈

Aliases

No aliases for this group.

Subgroups

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

790 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12496);