#493684

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1202 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)(5,6,9,15,10,14,24,29,11,13,23,28,8,21,22,34)(7,16,19,31,20,26,33,35,17,30,32,36,18,12,25,27);(1,2,3,4)(5,7,8,18,11,17,10,20)(6,12,14,16,13,26,21,30)(9,19,22,25,23,32,24,33)(15,27,29,31,28,35,34,36);(6,13)(9,23)(12,26)(14,21)(16,30)(19,32)(22,24)(25,33);(7,17)(12,26)(16,30)(18,20)(19,32)(25,33)(27,35)(31,36)

Polycyclic presentation:

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

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, 400880);