#494187

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

608 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)(5,6,8,15,9,16,22,31,10,14,21,30,23,13,28,29)(7,12,18,27,19,26,33,36,17,11,20,25,32,24,34,35);(5,7)(6,11)(8,20)(9,19)(10,17)(12,14)(13,26)(15,27)(16,24)(18,21)(22,34)(23,32)(25,30)(28,33)(29,35)(31,36);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,17)(11,14)(13,24)(15,27)(16,26)(20,21)(22,33)(23,32)(25,30)(28,34)(29,35)(31,36);(6,13,14,16)(7,17)(8,21)(11,24,12,26)(15,31,30,29)(19,32)(22,28)(25,36,27,35)

Polycyclic presentation:

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

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, 401383);