#494508

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1372 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=di,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320035, #72320788, #72320827
maximal normal subgroup of	#72320035, #72320788, #72320827

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401704);