#7659048

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,4,6)(3,5,7,8)(10,13)(11,15)(17,18)(19,25)(20,31)(21,27)(22,28)(23,24)(26,32)(29,33)(30,36)(34,44)(35,42)(37,38)(39,40)(41,45)(43,46)(47,48);(1,2)(3,5)(4,6)(7,8)(10,13)(14,16)(18,24)(19,27)(20,32)(28,38)(30,42)(33,39)(34,45)(46,47);(9,10)(11,14)(12,13)(15,16)(17,19,23,27)(18,25,24,21)(20,29,32,40)(22,30,37,42)(26,39,31,33)(28,36,38,35)(34,43,45,48)(41,47,44,46);(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,29)(21,33)(22,34)(23,32)(24,31)(25,39)(27,40)(28,41)(30,43)(35,46)(36,47)(37,45)(38,44)(42,48);(1,3,4,7)(2,5,6,8)(17,21,23,25)(18,27,24,19)(20,33,32,39)(22,36,37,35)(26,40,31,29)(28,30,38,42)(34,47,45,46)(41,43,44,48);(17,22)(18,28)(19,30)(20,34)(21,35)(23,37)(24,38)(25,36)(26,41)(27,42)(29,43)(31,44)(32,45)(33,46)(39,47)(40,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=h,e^2=g,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, 7566244);