#582204

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 489400);