#8756109

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

5152 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8663305);