#7715431

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

5242 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^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, 7622627);