#401303

order	512
nilpotency class	3
isoclinism	isoclinic class of #395148
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1259 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=fi,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, 308499);