#493085

order	512
nilpotency class	4
isoclinism	isoclinic class of #33469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1128 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)(5,6,10,14,12,15,11,29)(7,9,17,25,18,27,28,26)(8,13,22,32,20,30,23,34)(16,24,19,35,31,21,36,33);(2,4)(5,7,9,14,11,17,26,15,12,18,27,29,10,28,25,6)(8,19,21,13,23,31,35,32,20,36,24,30,22,16,33,34);(5,8,10,22,12,20,11,23)(6,13,14,32,15,30,29,34)(7,16,17,19,18,31,28,36)(9,24,25,35,27,21,26,33);(8,20)(13,30)(16,31)(19,36)(21,24)(22,23)(32,34)(33,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=i,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,c^b=ch,f^b=fi,d^c=di,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239482, #72239529, #72239554, …
maximal normal subgroup of	#72239482, #72239529, #72239554, …
maximal quotient of	#72319237, #72320014, C₂×#493085

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400281);