#7647224

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	#37845	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4423 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=gi,e^2=i,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, 7554420);