#123935

order	512
nilpotency class	3
isoclinism	isoclinic class of #8736
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,g^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fi,g^b=gi,h^c=hi

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, 31131);