#8105063

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303333
maximal normal subgroup of	#72303333
maximal quotient of	#72303424

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8012259);