#945522

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

1686 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gh,c^2=ghi,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 852718);