#7653028

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

4305 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)(11,18)(15,22)(19,23)(20,24)(25,26,30,34)(27,33,38,31)(28,32,39,29)(35,40,37,36);(1,2)(3,6)(4,7)(5,8)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,27)(26,31)(28,37)(29,36)(30,38)(32,40)(33,34)(35,39);(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,28)(26,32)(27,35)(29,34)(30,39)(31,36)(33,40)(37,38);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,29,30,32)(26,28,34,39)(27,36,38,40)(31,37,33,35);(1,3,5,4)(2,6,8,7)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,28)(26,32)(27,37)(29,34)(30,39)(31,40)(33,36)(35,38);(1,4,5,3)(2,7,8,6)(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,27)(26,33)(28,35)(29,36)(30,38)(31,34)(32,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7560224);