#497825

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
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	C₂²×D₄	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=de,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405021);