#7642164

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	#37276	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,e^2=hi,f^2=g,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, 7549360);