#1071554

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2588 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)(9,10,12,14)(11,15,16,13)(17,18)(19,20)(21,22)(23,24);(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,21)(22,24)(26,29)(30,32);(11,16)(13,15)(18,21)(19,24)(25,26)(27,30)(28,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,24)(22,23);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,21,22,18)(19,20,24,23)(25,27,28,31)(26,30,29,32)

Polycyclic presentation:

2^9|a^2=fg,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 978750);