#1215584

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

1724 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=fg,d^2=ghi,e^2=g,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, 1122780);