#7897863

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

4680 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7805059);