#8934370

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

5393 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340680
maximal normal subgroup of	#72340680

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841566);