#7922462

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

5329 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7829658);