#7642098

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	#36759	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4368 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)(12,20)(16,21)(18,24)(22,23)(25,26)(27,34)(28,36)(29,32)(30,31)(33,38)(35,39)(37,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(26,31)(32,34)(33,36)(39,40);(1,3,5,4)(2,7,6,8)(10,14)(15,17)(16,21)(22,23)(25,27,30,29)(26,32,31,34)(28,37,38,35)(33,40,36,39);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,30,38)(26,33,31,36)(27,35,29,37)(32,39,34,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,27)(26,34,31,32)(28,37,38,35)(33,40,36,39)

Polycyclic presentation:

2^9|c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549294);