#8078320

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

4885 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18,22,20)(19,25,26,23)(21,24,29,28)(27,31,32,30);(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,2,4,6)(3,5,7,8)(19,26)(23,25)(27,32)(30,31);(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,20,22,18)(19,23,26,25)(21,24,29,28)(27,31,32,30);(1,3,4,7)(2,5,6,8)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,2)(3,5)(4,6)(7,8)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7985516);