#7658451

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

4087 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)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(26,31)(27,35)(33,34)(36,37);(1,2)(3,6)(4,7)(5,8)(12,20)(15,22)(19,23)(21,24)(26,31)(32,38)(33,34)(39,40);(9,11,14,16)(10,13,17,18)(12,19,20,23)(15,21,22,24)(25,26,30,31)(27,32,35,38)(28,33,29,34)(36,39,37,40);(1,3,5,4)(2,6,8,7)(9,12)(10,15)(11,19)(13,21)(14,20)(16,23)(17,22)(18,24)(25,27)(26,32)(28,36)(29,37)(30,35)(31,38)(33,39)(34,40);(1,3,5,4)(2,6,8,7)(9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,22)(23,24)(25,28,30,29)(26,33,31,34)(27,36,35,37)(32,39,38,40);(1,4,5,3)(2,7,8,6)(9,13)(10,16)(11,17)(12,21)(14,18)(15,23)(19,22)(20,24)(25,29,30,28)(26,34,31,33)(27,37,35,36)(32,40,38,39)

Polycyclic presentation:

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

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, 7565647);