#392750

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1335 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dg,g^c=gi,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, 299946);