#7987306

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
prime radical	#36149	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4795 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=gi,f^2=ghi,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, 7894502);