#7986149

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

6723 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7893345);