#8372196

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

5010 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)(9,10,13,15)(11,12,16,14)(17,18,21,25)(19,23,27,22)(20,24,28,31)(26,30,32,29);(3,7)(6,8)(9,11,13,16)(10,14,15,12)(17,19,21,27)(18,22,25,23)(20,26,28,32)(24,29,31,30);(1,2,5,4)(3,6,7,8)(9,11)(10,12)(13,16)(14,15)(20,28)(24,31)(26,32)(29,30);(1,3,5,7)(2,6,4,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32);(1,4,5,2)(3,8,7,6)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(17,20,21,28)(18,24,25,31)(19,26,27,32)(22,29,23,30)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8279392);