#358051

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1734 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265247);