#7816034

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

4774 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,e^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, 7723230);