#7715484

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	#37659	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4868 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7622680);