#401059

order	512
nilpotency class	3
isoclinism	isoclinic class of #395148
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1724 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=efi,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308255);