#1409144

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

1942 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)(6,9)(7,11)(13,14,17,20)(15,19,16,18)(21,22,27,29)(23,30,24,28)(25,35,36,26)(31,32,33,34);(6,9)(10,12)(21,23,27,24)(22,28,29,30)(25,31,36,33)(26,32,35,34);(5,6,8,9)(7,10,11,12)(13,15,17,16)(14,18,20,19)(21,24,27,23)(22,28,29,30)(25,31,36,33)(26,34,35,32);(1,2,3,4)(21,25,27,36)(22,26,29,35)(23,31,24,33)(28,32,30,34);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,19,20,18)(21,26)(22,25)(23,32)(24,34)(27,35)(28,31)(29,36)(30,33)

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=fg,d^2=i,e^2=g,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, 1316340);