#412152

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319348);