#7630931

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4774 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,f^2=i,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, 7538127);