#7715520

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^2=h,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, 7622716);