#2168933

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1664 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=fh,e^2=gh,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, 2076129);