#6369927

order	512
nilpotency class	3
isoclinism	isoclinic class of #3585
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,f^2=i,g^2=i,h^2=i,b^a=bf,e^a=ei,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277123);