#7733670

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=gh,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, 7640866);