#7915507

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

4699 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=ghi,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, 7822703);