#7866437

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#38283	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,6)(7,8)(9,10)(11,20)(12,18)(13,22)(14,19)(15,17)(16,21)(23,24)(25,26,31,29)(27,35,37,32)(28,33,36,38)(30,34,40,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,14)(13,19)(15,21)(17,20)(18,24)(22,23)(25,27,31,37)(26,32,29,35)(28,30,36,40)(33,39,38,34);(10,17)(11,21)(12,22)(19,24)(25,26,31,29)(27,35,37,32)(28,33,36,38)(30,34,40,39);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,14)(13,17)(15,22)(16,24)(19,20)(21,23)(25,28,31,36)(26,33,29,38)(27,30,37,40)(32,39,35,34);(1,4,5,3)(2,8,6,7)(9,13,15,18)(10,12,17,22)(11,19,21,24)(14,20,23,16)(25,29,31,26)(27,35,37,32)(28,38,36,33)(30,34,40,39);(9,14)(10,19)(11,12)(13,16)(15,23)(17,24)(18,20)(21,22)(25,30)(26,34)(27,36)(28,37)(29,39)(31,40)(32,33)(35,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7773633);