#395171

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
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

1758 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=e,e^2=i,f^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,d^b=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302367);