#557151

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

1974 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)(5,6)(7,12)(8,11)(9,10)(13,14)(15,19)(16,17)(18,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(6,9)(11,12)(14,16)(18,19)(21,22,25,28)(23,29,31,26)(24,32,33,27)(30,34,36,35);(5,7,10,8)(6,11,9,12)(13,14,17,16)(15,18,20,19)(21,22)(23,26)(24,27)(25,28)(29,31)(30,34)(32,33)(35,36);(5,8,10,7)(6,12,9,11)(13,15)(14,18)(16,19)(17,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,9,10,6)(7,12,8,11)(13,16,17,14)(15,19,20,18)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 464347);