#7630665

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4704 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:

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

Polycyclic presentation:

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

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, 7537861);