#412727

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1100 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)(9,10,16,20,15,21,14,29)(11,19,13,28,24,35,23,17)(12,18,26,39,27,34,30,33)(22,32,36,31,37,40,25,38);(1,2,4,3)(5,6)(7,8)(9,11,14,23,15,24,16,13)(10,17,20,19,21,28,29,35)(12,22,26,36,27,37,30,25)(18,31,33,32,34,38,39,40);(1,3,4,2)(6,8)(9,12,14,26,15,27,16,30)(10,18,20,33,21,34,29,39)(11,22,23,36,24,37,13,25)(17,31,19,32,28,38,35,40);(9,13,16,24,15,23,14,11)(10,19,29,17,21,35,20,28)(12,25,30,37,27,36,26,22)(18,32,39,31,34,40,33,38)

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=e,d^2=hi,e^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,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, 319923);