#1409987

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
automorphism	?
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

1754 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)(5,6,8,10)(7,11,12,9)(21,22)(23,28)(24,32)(25,30)(26,27)(29,35)(31,36)(33,34);(13,14,17,16)(15,19,20,18)(22,27)(23,30)(29,32)(31,33);(5,6)(7,9)(8,10)(11,12)(13,15,17,20)(14,18,16,19)(21,23,26,30)(22,28,27,25)(24,33,35,31)(29,36,32,34);(1,2,3,4)(13,16,17,14)(15,19,20,18)(21,24)(22,29)(23,31)(25,36)(26,35)(27,32)(28,34)(30,33);(5,7,8,12)(6,9,10,11)(21,25,26,28)(22,30,27,23)(24,34,35,36)(29,31,32,33)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=h,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1317183);