#5608930

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2274 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

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, 5516126);