#208513

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ch,d^b=dh,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115709);