#705298

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2378 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=hi,d^2=fg,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, 612494);