#370958

order	512
nilpotency class	3
isoclinism	isoclinic class of #23900
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1350 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:

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

Polycyclic presentation:

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

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, 278154);