#7918531

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

4672 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=gh,e^2=i,f^2=ghi,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, 7825727);