#7715433

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×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 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)(10,13)(14,16)(17,18,22,23)(19,24,27,28)(20,31,33,25)(21,34,35,26)(29,41,39,36)(30,42,40,37)(32,38,45,44)(43,46,47,48);(9,10)(11,14)(12,13)(15,16)(18,23)(19,27)(25,31)(26,34)(29,39)(30,40)(38,44)(43,47);(2,5)(6,7)(17,18)(19,28)(20,31)(21,26)(22,23)(24,27)(25,33)(29,36)(30,42)(32,44)(34,35)(37,40)(38,45)(39,41)(43,46)(47,48);(1,2)(3,6)(4,5)(7,8)(17,19)(18,24)(20,29)(21,30)(22,27)(23,28)(25,36)(26,37)(31,41)(32,43)(33,39)(34,42)(35,40)(38,46)(44,48)(45,47);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,33)(18,25,23,31)(19,29,27,39)(21,32,35,45)(24,36,28,41)(26,38,34,44)(30,43,40,47)(37,46,42,48);(17,21,22,35)(18,26,23,34)(19,30,27,40)(20,32,33,45)(24,37,28,42)(25,38,31,44)(29,43,39,47)(36,46,41,48)

Polycyclic presentation:

2^9|a^2=i,e^2=g,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, 7622629);