#7763421

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

6638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=hi,e^2=i,f^2=i,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, 7670617);