#7657939

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4164 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=hi,e^2=gi,f^2=gi,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, 7565135);