#381172

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1214 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=gi,f^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288368);