#7646993

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4283 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=hi,e^2=ghi,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, 7554189);