#7646008

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

7140 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)(9,17)(13,20)(15,19)(16,18)(21,22,26,30)(23,31,33,27)(24,34,32,28)(25,29,35,36);(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,18)(16,19)(17,20)(21,23,26,33)(22,27,30,31)(24,25,32,35)(28,29,34,36);(5,7)(6,8)(9,15)(10,12)(11,14)(13,16)(17,19)(18,20)(21,23,26,33)(22,27,30,31)(24,25,32,35)(28,29,34,36);(5,7)(6,11)(8,14)(9,15)(10,12)(13,18)(16,20)(17,19)(21,24)(22,28)(23,25)(26,32)(27,29)(30,34)(31,36)(33,35);(5,8,10,11)(6,12,14,7)(9,16,17,18)(13,19,20,15)(21,25)(22,29)(23,32)(24,33)(26,35)(27,34)(28,31)(30,36);(5,9)(6,13)(7,15)(8,16)(10,17)(11,18)(12,19)(14,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,33)(32,34)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,e^2=i,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, 7553204);