#8934371

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

5425 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)(17,18,22,26)(19,23,24,20)(21,25,30,32)(27,31,29,28);(1,3,5,8)(2,6,7,4)(9,11)(10,12)(13,16)(14,15)(17,19,22,24)(18,20,26,23)(21,27,30,29)(25,28,32,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,24)(18,23,26,20)(21,29,30,27)(25,28,32,31);(2,7)(3,8)(10,14)(11,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,32)(29,30);(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,29,32,27);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)

Polycyclic presentation:

2^9|a^2=gi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841567);