#8874931

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

5558 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261791, #72261948
maximal normal subgroup of	#72261791, #72261948

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8782127);