#8418201

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

5117 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)(19,24,29,31)(20,21,26,25)(27,32,30,28);(3,7)(6,8)(10,11)(14,15)(18,23)(20,26)(24,31)(27,30);(1,2,5,4)(3,6,7,8)(9,11,13,10)(12,15,16,14)(18,23)(21,25)(24,31)(28,32);(1,3,5,7)(2,6,4,8)(9,12)(10,14)(11,15)(13,16)(17,19,22,29)(18,24,23,31)(20,30,26,27)(21,28,25,32);(1,4,5,2)(3,8,7,6)(17,20,22,26)(18,25,23,21)(19,27,29,30)(24,28,31,32);(17,21,22,25)(18,26,23,20)(19,28,29,32)(24,27,31,30)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8325397);