#467104

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1188 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)(5,6,8,10)(7,9,11,12)(14,17)(18,20)(21,22,26,33,25,29,24,28)(23,27,34,36,31,30,32,35);(5,7,8,11)(6,9,10,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,32)(25,31)(26,34)(28,35)(29,30)(33,36);(7,11)(9,12)(14,17)(15,19)(21,22,25,29)(23,30,31,27)(24,33,26,28)(32,35,34,36);(2,4)(13,15,16,19)(14,18,17,20)(23,31)(27,30)(32,34)(35,36)

Polycyclic presentation:

2^9|a^2=efg,b^2=egi,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,f^c=fi

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, 374300);