#4416487

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1802 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=i,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4323683);