#495016

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

692 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402212);