#1079013

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2723 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)(3,5,7,8)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,23,24,21)(26,29)(27,31);(9,10)(11,13)(12,14)(15,16)(17,19,20,24)(18,21,22,23)(25,26)(27,30)(28,29)(31,32);(11,16)(13,15)(18,22)(21,23)(26,29)(30,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,21)(20,24)(22,23)(25,27,28,31)(26,30,29,32);(1,3,4,7)(2,5,6,8)(9,11,12,16)(10,13,14,15)(17,18)(19,21)(20,22)(23,24)(25,26)(27,30)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,e^2=ghi,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.

Relationships

maximal subgroup of	#72222757, #72222847
maximal normal subgroup of	#72222757, #72222847
maximal quotient of	#72222738

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 986209);