#2315707

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1738 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)(5,6,8,9)(7,11,12,10)(13,14,17,16)(15,18,19,20)(21,22)(23,30)(24,25)(26,28)(27,29)(31,35)(32,33)(34,36);(6,9)(10,11)(15,19)(18,20)(21,23,27,24)(22,25,29,30)(26,31,34,32)(28,33,36,35);(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,20)(17,19)(21,24,27,23)(22,25,29,30)(26,31,34,32)(28,35,36,33);(1,2,3,4)(21,25)(22,23)(24,29)(26,33)(27,30)(28,31)(32,36)(34,35);(5,7,8,12)(6,10,9,11)(13,16,17,14)(15,20,19,18)(21,26,27,34)(22,28,29,36)(23,31,24,32)(25,33,30,35)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=fg,d^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 2222903);