#7657122

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	#40535	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4383 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)(11,17)(12,16)(13,18)(14,19)(15,22)(20,21)(23,24)(26,32)(28,35)(33,36)(37,39);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24)(25,26)(27,36)(28,34)(29,33)(30,35)(31,32)(37,38)(39,40);(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24)(25,27,31,29)(26,33,32,36)(28,37,35,39)(30,38,34,40);(1,3,5,4)(2,7,6,8)(9,12,14,18)(10,16,19,13)(11,20,22,24)(15,23,17,21)(25,28,31,35)(26,34,32,30)(27,37,29,39)(33,40,36,38);(1,3,5,4)(2,7,6,8)(9,11)(10,17)(12,20)(13,23)(14,22)(15,19)(16,21)(18,24)(25,29,31,27)(26,33,32,36)(28,39,35,37)(30,38,34,40);(1,4,5,3)(2,8,6,7)(9,13,14,16)(10,18,19,12)(11,21,22,23)(15,24,17,20)(25,30,31,34)(26,35,32,28)(27,38,29,40)(33,39,36,37)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7564318);