#7966554

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

4912 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)(3,5)(4,7)(6,8)(10,14)(15,16)(17,18)(19,22)(20,28)(21,31)(23,25)(24,30)(26,32)(27,29);(1,3)(2,5)(4,8)(6,7)(9,10,13,14)(11,15,12,16)(17,19,23,21)(18,22,25,31)(20,27,30,26)(24,32,28,29);(1,2)(3,6)(4,7)(5,8)(17,20,23,30)(18,24,25,28)(19,26,21,27)(22,29,31,32);(1,3)(2,6)(4,8)(5,7)(17,21,23,19)(18,22,25,31)(20,26,30,27)(24,32,28,29);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,27)(21,26)(22,32)(23,30)(25,28)(29,31);(1,2)(3,6)(4,7)(5,8)(9,12,13,11)(10,16,14,15)(17,22)(18,19)(20,29)(21,25)(23,31)(24,26)(27,28)(30,32)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,e^2=g,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, 7873750);