#4416939

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1890 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=h,d^2=fhi,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, 4324135);