#5622650

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2300 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22,25,29)(23,27,31,26)(24,28,33,32)(30,35,36,34);(7,11)(10,12)(13,14)(15,19)(16,17)(18,20)(21,23,25,31)(22,26,29,27)(24,30,33,36)(28,34,32,35);(6,9)(10,12)(14,17)(18,19)(21,23)(22,27)(24,30)(25,31)(26,29)(28,35)(32,34)(33,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,32)(25,29)(27,31)(28,33)(30,35)(34,36);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,20)(17,19)(21,24)(22,28)(23,30)(25,33)(26,34)(27,35)(29,32)(31,36)

Polycyclic presentation:

2^9|a^2=f,b^2=f,e^2=fgh,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, 5529846);