#5363296

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 1048576
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3115 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300717, #72300749, #72300763
maximal normal subgroup of	#72300717, #72300749, #72300763

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5270492);