#7733524

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

4824 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7640720);