#7642721

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

7086 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:

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

Polycyclic presentation:

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

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, 7549917);