#392224

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299420);