#366012

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,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, 273208);