#7630820

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4696 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538016);