#8757260

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

4137 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)(3,8)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(19,25)(22,26)(27,31)(30,32)(34,36)(35,41)(37,45)(46,48);(1,3,7,4)(2,8,9,5)(6,12,15,11)(10,13,14,16)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,30)(31,32)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(2,9)(3,4)(10,14)(11,12)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(1,4,7,3)(2,5,9,8)(6,12,15,11)(10,13,14,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,5)(2,3)(4,9)(6,13)(7,8)(10,11)(12,14)(15,16)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,6,7,15)(2,10,9,14)(3,11,4,12)(5,13,8,16)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=h,e^2=hi,f^2=gh,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, 8664456);