#642942

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

2374 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)(7,11)(10,12)(21,22,26,24)(23,28,31,27)(25,32,33,29)(30,34,36,35);(6,9)(10,12)(13,14,16,18)(15,17,19,20)(21,23,26,31)(22,27,24,28)(25,30,33,36)(29,34,32,35);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,28)(24,27)(25,30)(26,31)(29,35)(32,34)(33,36);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,17,18,20)(21,24,26,22)(23,28,31,27)(25,32,33,29)(30,35,36,34);(21,25,26,33)(22,29,24,32)(23,30,31,36)(27,34,28,35)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=g,d^2=hi,e^2=f,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, 550138);