#8095352

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	#52972	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,8)(4,5)(6,7)(9,13)(10,14)(11,12)(15,16)(18,21)(19,25)(24,29)(26,31)(33,34)(35,38)(36,39)(37,42)(40,45)(41,44)(43,47)(46,48);(3,10)(8,14)(11,15)(12,16)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(34,35)(39,41)(40,42)(46,47);(2,7)(4,9)(8,14)(11,15)(20,28)(24,29)(26,31)(30,32)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,3,6,10)(2,8,7,14)(4,11,9,15)(5,12,13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,6,9)(2,5,7,13)(3,11,10,15)(8,12,14,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,5)(2,9)(3,12)(4,7)(6,13)(8,15)(10,16)(11,14)(17,21,22,18)(19,27,25,23)(20,29,28,24)(26,32,31,30)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002548);