#588401

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

1846 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)(6,7)(10,11)(13,14)(15,19)(16,20)(17,18)(21,22,26,29)(23,30,27,24)(25,33,34,28)(31,32,35,36);(1,3,4,2)(5,6,9,7)(8,11,12,10)(13,15,17,20)(14,16,18,19)(23,27)(24,30)(31,35)(32,36);(6,7)(10,11)(14,18)(15,20)(21,23,26,27)(22,24,29,30)(25,31,34,35)(28,32,33,36);(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,20)(18,19)(21,24)(22,27)(23,29)(25,32)(26,30)(28,35)(31,33)(34,36);(5,8,9,12)(6,10,7,11)(13,16)(14,15)(17,19)(18,20)(21,25,26,34)(22,28,29,33)(23,31,27,35)(24,32,30,36)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=fh,d^2=gh,e^2=fg,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, 495597);