#8898048

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

4131 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,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, 8805244);