#7743817

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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

6320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

(2,4)(5,6)(7,11)(8,13)(9,14)(10,15)(12,16)(17,18)(19,20)(21,22,27,29)(23,24,25,28)(26,30,34,36)(31,32,33,35);(1,2,3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,15)(17,19)(18,20)(21,23,27,25)(22,28,29,24)(26,33,34,31)(30,32,36,35);(7,16)(8,18)(11,12)(13,17)(22,29)(23,25)(30,36)(31,33);(5,7)(6,12)(8,14)(9,17)(10,16)(11,15)(13,19)(18,20)(21,24)(22,23)(25,29)(26,32)(27,28)(30,31)(33,36)(34,35);(5,8)(6,13)(7,14)(9,11)(10,18)(12,19)(15,17)(16,20)(21,25,27,23)(22,24,29,28)(26,33,34,31)(30,32,36,35);(5,9,10,19)(6,14,15,20)(7,17,16,13)(8,11,18,12)(21,26,27,34)(22,30,29,36)(23,31,25,33)(24,32,28,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7651013);