#8073212

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	#51419	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4400 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)(10,14)(15,17)(16,22)(23,24)(25,26,31,34)(27,29,32,28)(30,33,38,40)(35,37,39,36);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,27,31,32)(26,28,34,29)(30,35,38,39)(33,36,40,37);(10,14)(15,17)(16,22)(23,24)(25,28,31,29)(26,32,34,27)(30,37,38,36)(33,35,40,39);(1,3,5,4)(2,7,6,8)(9,11,13,19)(10,15,14,17)(12,20,21,18)(16,24,22,23)(25,29,31,28)(26,32,34,27)(30,36,38,37)(33,35,40,39);(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,30,31,38)(26,33,34,40)(27,35,32,39)(28,36,29,37)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7980408);