#493061

order	512
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=ghi,h^a=hi,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323138
maximal normal subgroup of	#72323138
maximal quotient of	#72322679

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400257);