#369916

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

1092 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)(15,19)(16,18)(17,20)(21,22)(23,29)(24,26)(25,28)(27,31)(30,34)(32,33)(35,36);(1,3,4,2)(5,7,9,8)(6,10,12,11)(13,15,17,16)(14,18,20,19)(22,26)(23,30)(25,32)(27,34)(29,35)(31,36);(5,7,9,8)(6,11,12,10)(13,16,17,15)(14,19,20,18)(21,23,25,30,24,31,32,36)(22,27,28,35,26,29,33,34);(5,8,9,7)(6,11,12,10)(14,20)(18,19)(22,26)(27,29)(28,33)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=egi,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277112);