#7763424

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

6930 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)(11,14)(15,19)(16,17)(18,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,35)(32,36)(33,34);(1,2)(3,4)(6,10)(7,14)(12,16)(15,20)(21,22,26,30)(23,28,24,27)(25,29,34,33)(31,36,32,35);(5,6,9,10)(7,11,14,13)(8,16,17,12)(15,19,20,18)(21,22)(23,27)(24,28)(25,33)(26,30)(29,34)(31,36)(32,35);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,23,26,24)(22,27,30,28)(25,31,34,32)(29,35,33,36);(5,7,9,14)(6,11,10,13)(8,15,17,20)(12,18,16,19)(21,24,26,23)(22,28,30,27)(25,32,34,31)(29,36,33,35);(5,8,9,17)(6,12,10,16)(7,15,14,20)(11,18,13,19)(21,25,26,34)(22,29,30,33)(23,31,24,32)(27,35,28,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,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, 7670620);