#2586462

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,d^2=hi,e^2=hi,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, 2493658);