#7657237

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4160 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)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(26,31)(27,35)(33,34)(36,37);(1,2)(3,7)(4,8)(5,6)(9,11,14,21)(10,15,18,19)(12,13,20,23)(16,22,24,17)(26,31)(32,38)(33,34)(39,40);(9,11,14,21)(10,15,18,19)(12,13,20,23)(16,22,24,17)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(9,12,14,20)(10,16,18,24)(11,13,21,23)(15,22,19,17)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,39)(34,40);(1,3,5,4)(2,7,6,8)(9,13)(10,17)(11,20)(12,21)(14,23)(15,16)(18,22)(19,24)(25,28,30,29)(26,33,31,34)(27,36,35,37)(32,39,38,40);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,16)(13,22)(14,18)(17,23)(19,21)(20,24)(25,29,30,28)(26,34,31,33)(27,37,35,36)(32,40,38,39)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=g,e^2=gh,f^2=gh,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, 7564433);