#8838479

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

4241 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,16)(14,15)(18,20)(19,25)(24,28)(26,31)(33,34)(35,39)(36,40)(37,46)(38,42)(41,47)(43,44)(45,48);(2,7)(4,13)(5,8)(11,14)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30)(33,35)(34,39)(36,37)(38,43)(40,46)(41,48)(42,44)(45,47);(2,7)(3,10)(9,12)(11,14)(19,25)(23,27)(26,31)(30,32)(35,43)(36,45)(39,44)(40,48);(1,3,6,10)(2,8,7,5)(4,14,13,11)(9,15,12,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36,38,45)(34,40,42,48)(35,37,43,47)(39,46,44,41);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,37)(34,41)(35,36)(38,47)(39,48)(40,44)(42,46)(43,45);(1,5,6,8)(2,10,7,3)(4,15,13,16)(9,14,12,11)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,48)(46,47)

Polycyclic presentation:

2^9|b^2=gh,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8745675);