#7747299

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

4642 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)(9,10,15,14)(11,17,12,16)(13,21,22,18)(19,23,20,24)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,16,14,17)(13,20,22,19)(18,24,21,23)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,3,5,4)(2,7,6,8)(10,14)(11,12)(18,21)(19,20)(26,28)(32,33)(34,36)(39,40);(9,12,15,11)(10,17,14,16)(13,20,22,19)(18,24,21,23)(25,28,31,26)(27,33,29,32)(30,36,38,34)(35,40,37,39);(1,3,5,4)(2,7,6,8)(9,13,15,22)(10,18,14,21)(11,19,12,20)(16,23,17,24)(25,29,31,27)(26,33,28,32)(30,37,38,35)(34,40,36,39);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,17,12,16)(13,21,22,18)(19,24,20,23)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7654495);