#8905176

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

4065 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)(6,14)(7,9)(10,15)(11,13)(12,16)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,31,28)(33,34)(35,38)(36,41)(37,43)(39,44)(40,47)(42,46)(45,48);(1,3,7,4)(2,8,9,5)(6,11,15,12)(10,16,14,13)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(33,35)(34,40)(36,45)(37,46)(38,44)(39,47)(41,43)(42,48);(2,9)(3,4)(6,15)(13,16)(19,27)(20,29)(23,30)(24,32)(33,36,39,46)(34,41,44,42)(35,45,47,37)(38,48,40,43);(1,4,7,3)(2,5,9,8)(6,12,15,11)(10,13,14,16)(17,20)(18,24)(19,28)(21,23)(22,29)(25,27)(26,32)(30,31)(33,36)(34,42)(35,45)(37,47)(38,43)(39,46)(40,48)(41,44);(1,5)(2,4)(3,9)(6,13)(7,8)(10,12)(11,14)(15,16)(17,21)(18,25)(19,24)(20,30)(22,31)(23,29)(26,28)(27,32)(33,37,39,45)(34,43,44,48)(35,46,47,36)(38,42,40,41);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,16)(9,14)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(33,38,39,40)(34,35,44,47)(36,43,46,48)(37,42,45,41)

Polycyclic presentation:

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

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, 8812372);