#641386

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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

2029 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)(6,10)(7,12)(14,18)(19,20)(21,22,27,30)(23,31,24,28)(25,36,29,26)(32,33,34,35);(14,18)(19,20)(21,23,27,24)(22,28,30,31)(25,32,29,34)(26,33,36,35);(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,20)(17,18)(21,24,27,23)(22,28,30,31)(25,32,29,34)(26,35,36,33);(1,2,3,4)(5,7,9,12)(6,11,10,8)(13,15,17,16)(14,19,18,20)(21,25,27,29)(22,26,30,36)(23,32,24,34)(28,33,31,35);(5,8,9,11)(6,12,10,7)(13,16,17,15)(14,20,18,19)(21,26)(22,29)(23,33)(24,35)(25,30)(27,36)(28,34)(31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 548582);