#7736712

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

4937 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:

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

Polycyclic presentation:

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

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, 7643908);