#384990

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

1028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=dei,d^c=di

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, 292186);