#600725

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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

1722 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)(8,10)(9,12)(15,19)(18,20)(21,22)(23,28)(24,29)(25,30)(26,27)(31,36)(32,33)(34,35);(5,6,8,10)(7,11,12,9)(14,17)(18,20)(22,27)(25,28)(29,35)(33,36);(1,2,4,3)(5,6,8,10)(7,9,12,11)(13,14)(15,18)(16,17)(19,20)(21,23,26,30)(22,28,27,25)(24,32,34,31)(29,33,35,36);(5,6)(7,9)(8,10)(11,12)(13,15,16,19)(14,18,17,20)(21,24,26,34)(22,29,27,35)(23,31,30,32)(25,33,28,36);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(21,25,26,28)(22,30,27,23)(24,33,34,36)(29,32,35,31)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 507921);