#7752429

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

4898 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)(17,18,23,22)(19,27,28,24)(20,25,29,31)(21,32,30,26);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19,23,28)(18,24,22,27)(20,30,29,21)(25,32,31,26);(2,5)(6,7)(9,11,13,12)(10,15,14,16)(18,22)(19,28)(20,29)(26,32);(1,3)(2,6)(4,8)(5,7)(17,20,23,29)(18,25,22,31)(19,21,28,30)(24,26,27,32);(1,3)(2,6)(4,8)(5,7)(17,21)(18,26)(19,29)(20,28)(22,32)(23,30)(24,31)(25,27);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,16,14,15)(17,22,23,18)(19,27,28,24)(20,31,29,25)(21,32,30,26)

Polycyclic presentation:

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

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, 7659625);