#9054387

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4401 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961583);