#1142624

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

1900 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:

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

Polycyclic presentation:

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

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, 1049820);