#7737502

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

4792 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)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,2)(3,8)(4,5)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(2,7)(3,4)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31);(1,3,6,4)(2,8,7,5)(9,11,13,12)(10,15,14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32);(1,4,6,3)(2,5,7,8)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(1,5)(2,3)(4,7)(6,8)(9,12,13,11)(10,16,14,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)

Polycyclic presentation:

2^9|c^2=h,d^2=ghi,e^2=i,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, 7644698);