#7659044

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

6712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7566240);