#2594779

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fg,c^2=fghi,d^2=fi,e^2=ghi,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, 2501975);