#7733596

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

5281 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,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, 7640792);