#356139

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1348 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263335);