#7736419

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

4940 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=ghi,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, 7643615);