#8772875

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	#36318	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4412 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8680071);