#946127

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

1770 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)(7,11)(9,12)(13,14,17,19)(15,16,18,20)(21,22,25,26)(23,29,30,27)(24,32,33,28)(31,34,36,35);(5,6,8,10)(7,12,11,9)(13,15)(14,16)(17,18)(19,20)(22,26)(23,30)(28,32)(31,36);(5,7,8,11)(6,9,10,12)(13,16)(14,18)(15,19)(17,20)(21,23,25,30)(22,27,26,29)(24,31,33,36)(28,34,32,35);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36);(21,24,25,33)(22,28,26,32)(23,31,30,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=g,d^2=fi,e^2=f,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, 853323);