#7715552

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

4310 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,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, 7622748);