#8755052

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

3975 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)(4,6)(5,14)(7,9)(10,13)(11,16)(12,15)(17,18)(19,25)(20,27)(21,29)(22,24)(23,30)(26,28)(31,32)(34,36)(35,38)(37,45)(47,48);(1,3)(2,8)(4,13)(5,11)(6,10)(7,12)(9,15)(14,16)(17,18,22,24)(19,25,27,20)(21,23,30,29)(26,31,32,28)(33,34,39,36)(35,40,38,43)(37,41,45,44)(42,47,46,48);(2,9)(3,12)(4,11)(10,14)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32)(33,35,39,38)(34,40,36,43)(37,42,45,46)(41,47,44,48);(1,4,7,11)(2,6,9,16)(3,13,12,5)(8,10,15,14)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,36,39,34)(35,40,38,43)(37,44,45,41)(42,47,46,48);(1,5,7,13)(2,10,9,14)(3,11,12,4)(6,15,16,8)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48);(1,6)(2,11)(3,14)(4,9)(5,15)(7,16)(8,13)(10,12)(17,21,22,30)(18,23,24,29)(19,26,27,32)(20,28,25,31)(33,38,39,35)(34,40,36,43)(37,46,45,42)(41,47,44,48)

Polycyclic presentation:

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

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, 8662248);