#8600109

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

5649 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261815, #72261985
maximal normal subgroup of	#72261815, #72261985

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8507305);