#7987960

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

4756 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=gi,f^2=i,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, 7895156);