#7645841

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

6648 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)(7,12)(8,16)(9,10)(11,14)(13,19)(15,18)(17,20)(22,26)(23,29)(28,33)(30,35);(1,2)(3,4)(5,7,11,9)(6,12,14,10)(8,15,19,17)(13,20,16,18)(22,26)(27,32)(28,33)(34,36);(5,8)(6,13)(7,15)(9,17)(10,18)(11,19)(12,20)(14,16)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(5,9,11,7)(6,12,14,10)(8,17,19,15)(13,20,16,18)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,10,11,12)(6,9,14,7)(8,18,19,20)(13,17,16,15)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36);(5,8)(6,13)(7,15)(9,17)(10,18)(11,19)(12,20)(14,16)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553037);