#2568621

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1808 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=i,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, 2475817);