#7746549

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

4773 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:

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

Polycyclic presentation:

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

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, 7653745);