#394812

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1607 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^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,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, 302008);