#8755327

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

4125 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(18,22)(19,25)(24,27)(26,31)(34,36)(35,41)(37,45)(46,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(2,6)(7,9)(8,12)(14,15)(19,25)(23,29)(26,31)(30,32)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=hi,e^2=h,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, 8662523);