#2329233

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1732 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=fgh,d^2=hi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2236429);