#355861

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

1038 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=fg,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cei,d^b=di,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263057);