#392747

order	512
nilpotency class	3
isoclinism	isoclinic class of #32888
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1307 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)(9,10,16,20,14,19,15,30)(11,21,24,32,23,13,18,29)(12,25,28,17,27,35,31,34)(22,37,39,36,38,40,33,26);(2,5)(6,8)(9,11,16,24,14,23,15,18)(10,13,20,29,19,21,30,32)(12,22,28,39,27,38,31,33)(17,26,35,37,34,36,25,40);(1,3,4,7)(2,6,5,8)(9,12,14,27)(10,17,19,34)(11,22,23,38)(13,26,21,36)(15,31,16,28)(18,33,24,39)(20,35,30,25)(29,37,32,40);(9,13)(10,18)(11,20)(12,26)(14,21)(15,32)(16,29)(17,33)(19,24)(22,35)(23,30)(25,38)(27,36)(28,37)(31,40)(34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,f^c=fi,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, 299943);