#370868

order	512
nilpotency class	3
isoclinism	isoclinic class of #23900
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

1110 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,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, 278064);