#8898050

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

4211 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805246);