#7645812

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

4152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7553008);