#476694

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1394 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ei,c^2=i,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383890);