#7916658

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4654 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)(11,19)(12,23)(13,18)(14,16)(15,22)(17,21)(20,24)(25,26,31,33)(27,30,35,32)(28,38,34,29)(36,37,40,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23)(25,27)(26,32)(28,36)(29,37)(30,33)(31,35)(34,40)(38,39);(1,3,5,4)(2,7,6,8)(10,16)(15,19)(17,23)(18,24)(26,33)(27,35)(28,34)(37,39);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,28,31,34)(26,29,33,38)(27,36,35,40)(30,39,32,37);(1,4,5,3)(2,8,6,7)(9,12,14,21)(10,17,16,23)(11,20,22,13)(15,24,19,18)(25,29)(26,34)(27,37)(28,33)(30,36)(31,38)(32,40)(35,39);(9,13,14,20)(10,18,16,24)(11,21,22,12)(15,23,19,17)(25,30)(26,35)(27,33)(28,39)(29,40)(31,32)(34,37)(36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7823854);