#4380618

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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

1878 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)(13,14,17,18)(15,16,20,19)(21,22,27,30)(23,28,26,36)(24,33,29,25)(31,32,34,35);(6,7)(10,11)(14,18)(15,20)(21,23,27,26)(22,28,30,36)(24,34,29,31)(25,32,33,35);(8,12)(10,11)(14,18)(16,19)(21,24,27,29)(22,25,30,33)(23,31,26,34)(28,35,36,32);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,20)(14,19,18,16)(21,25)(22,29)(23,32)(24,30)(26,35)(27,33)(28,31)(34,36);(5,8)(6,10)(7,11)(9,12)(13,16,17,19)(14,20,18,15)(21,26,27,23)(22,28,30,36)(24,34,29,31)(25,35,33,32)

Polycyclic presentation:

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

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, 4287814);