#7674429

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

4314 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581625);