#4467787

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
automorphism	Aut(G) order 1048576
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

1674 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)(5,6)(7,9)(8,10)(11,12)(13,14,18,19)(15,20,17,16)(21,22,25,28)(23,26,30,33)(24,27,31,34)(29,32,35,36);(6,7)(8,12)(14,19)(15,17)(21,23,25,30)(22,26,28,33)(24,29,31,35)(27,32,34,36);(6,7)(10,11)(13,15,18,17)(14,20,19,16)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,20)(14,15,19,17)(21,23)(22,26)(24,29)(25,30)(27,32)(28,33)(31,35)(34,36);(5,8)(6,10)(7,11)(9,12)(13,17,18,15)(14,20,19,16)(21,24,25,31)(22,27,28,34)(23,29,30,35)(26,32,33,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=h,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4374983);