#7973961

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

4960 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7881157);