#546008

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fgi,d^2=gh,e^2=fh,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 453204);