#413068

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1602 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)(5,6,8,7)(9,10,16,21)(11,22,26,36)(12,27,29,38)(13,30,28,40)(14,32,15,20)(17,33,37,25)(18,34,35,24)(19,23,39,31);(1,2,4,3)(5,7,8,6)(9,11,15,33,16,26,14,25)(10,17,32,36,21,37,20,22)(12,28,34,31,29,13,24,23)(18,30,38,19,35,40,27,39);(1,3,4,2)(6,7)(9,12,14,24,16,29,15,34)(10,18,20,27,21,35,32,38)(11,23,25,13,26,31,33,28)(17,19,22,30,37,39,36,40);(9,13,15,23,16,28,14,31)(10,19,32,40,21,39,20,30)(11,24,33,12,26,34,25,29)(17,35,36,27,37,18,22,38)

Polycyclic presentation:

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

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, 320264);