#698660

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
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

2370 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=f,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 605856);