#720342

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

2078 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)(5,6,10,9)(7,12,11,8)(13,14,17,16)(15,19,20,18)(21,22)(23,26)(24,28)(25,33)(27,30)(29,35)(31,36)(32,34);(15,20)(18,19)(21,23,27,24)(22,28,30,26)(25,31,35,32)(29,36,33,34);(5,7)(6,8)(9,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,24,27,23)(22,28,30,26)(25,31,35,32)(29,34,33,36);(1,2,3,4)(5,8,10,12)(6,11,9,7)(13,14,17,16)(15,18,20,19)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,36)(30,33);(5,9,10,6)(7,12,11,8)(13,16,17,14)(15,19,20,18)(21,26,27,28)(22,23,30,24)(25,34,35,36)(29,31,33,32)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=fhi,e^2=fg,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, 627538);