#4319972

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2090 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)(5,6,9,11)(7,12,8,10)(14,17)(15,19)(21,22,26,24)(23,27,29,30)(25,32,33,28)(31,36,35,34);(5,7,9,8)(6,10,11,12)(13,14)(15,20)(16,17)(18,19)(23,29)(27,30)(31,35)(34,36);(7,8)(10,12)(13,14,16,17)(15,18,19,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34);(5,6)(7,12)(8,10)(9,11)(13,15,16,19)(14,18,17,20)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=g,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4227168);