#375982

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1254 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)(11,17,23,35,24,36,31,34)(12,25,28,29,27,18,32,13)(21,33,38,37,39,22,26,40);(2,5)(6,8)(9,11)(10,17)(12,26)(13,22)(14,24)(15,31)(16,23)(18,37)(19,36)(20,35)(21,28)(25,40)(27,38)(29,33)(30,34)(32,39);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,21)(13,20)(14,27)(15,32)(16,28)(17,33)(19,25)(22,36)(23,38)(24,39)(26,31)(29,30)(34,40)(35,37);(2,5)(6,8)(9,13,16,25,14,29,15,18)(10,12,20,28,19,27,30,32)(11,22,23,40,24,33,31,37)(17,26,35,21,36,38,34,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283178);