#1127447

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1824 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:

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

Polycyclic presentation:

2^9|b^2=fghi,c^2=fgh,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

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, 1034643);