#739734

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

2052 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fi,d^2=fi,e^2=g,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, 646930);