#6132602

order	512
nilpotency class	2
isoclinism	isoclinic class of #6131635
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1964 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg

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, 6039798);