#597209

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fghi,d^2=hi,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, 504405);