#7812022

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

5075 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7719218);