#8197815

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

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

5423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263771
maximal normal subgroup of	#72263771
maximal quotient of	#72263776

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8105011);