#7719086

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

6558 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)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,22)(23,28)(24,30)(25,27)(26,36)(29,34)(31,32)(33,35);(1,2)(3,4)(21,23,27,33)(22,28,25,35)(24,32,34,26)(29,36,30,31);(6,10)(11,13)(12,17)(18,19)(21,24,27,34)(22,29,25,30)(23,26,33,32)(28,31,35,36);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,24)(22,30)(23,32)(25,29)(26,33)(27,34)(28,31)(35,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,25,27,22)(23,28,33,35)(24,29,34,30)(26,36,32,31);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,26,27,32)(22,31,25,36)(23,24,33,34)(28,29,35,30)

Polycyclic presentation:

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

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, 7626282);