#8850658

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

3952 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=hi,e^2=g,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, 8757854);