#7763439

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

7012 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)(5,6,11,12)(7,9,8,13)(10,14,18,19)(15,17,16,20)(21,22)(23,27)(24,32)(25,30)(26,28)(29,34)(31,36)(33,35);(1,2)(3,4)(5,7)(6,9)(8,11)(10,15)(12,13)(14,17)(16,18)(19,20)(21,23,26,25)(22,27,28,30)(24,31,34,33)(29,35,32,36);(6,12)(7,8)(10,18)(17,20)(22,28)(27,30)(29,32)(35,36);(5,8,11,7)(6,13,12,9)(10,16,18,15)(14,20,19,17)(21,22)(23,27)(24,29)(25,30)(26,28)(31,35)(32,34)(33,36);(5,9,11,13)(6,7,12,8)(10,17,18,20)(14,15,19,16)(21,24)(22,29)(23,31)(25,33)(26,34)(27,35)(28,32)(30,36);(5,10)(6,14)(7,15)(8,16)(9,17)(11,18)(12,19)(13,20)(21,25,26,23)(22,30,28,27)(24,33,34,31)(29,36,32,35)

Polycyclic presentation:

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

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, 7670635);