#1085083

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2860 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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=fh,e^2=fg,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, 992279);