#4315196

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

2068 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)(5,6,8,10)(7,11,12,9)(13,14,17,16)(15,19,20,18)(21,22,26,29)(23,30,27,24)(25,33,35,28)(31,32,34,36);(7,12)(9,11)(21,23,26,27)(22,24,29,30)(25,34,35,31)(28,36,33,32);(5,7)(6,9)(8,12)(10,11)(13,15,17,20)(14,18,16,19)(21,23)(22,24)(25,31)(26,27)(28,32)(29,30)(33,36)(34,35);(1,2)(3,4)(13,16,17,14)(15,19,20,18)(21,24)(22,27)(23,29)(25,32)(26,30)(28,34)(31,33)(35,36);(5,6)(7,9)(8,10)(11,12)(21,25,26,35)(22,28,29,33)(23,31,27,34)(24,32,30,36)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gh,d^2=gh,e^2=f,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, 4222392);