#8143153

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3833 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)(3,13)(4,10)(5,6)(7,15)(8,12)(9,16)(11,14)(17,18,23,26)(19,21,22,24)(20,28,31,25)(27,32,30,29)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47);(1,3)(2,9)(4,7)(5,14)(6,15)(8,16)(10,11)(12,13)(17,19,23,22)(18,24,26,21)(20,27,31,30)(25,32,28,29)(33,34,38,42)(35,39,46,43)(36,40,37,41)(44,47,45,48);(1,4,8,5)(2,10,12,6)(3,7,16,14)(9,11,13,15)(17,20)(18,25)(19,27)(21,29)(22,30)(23,31)(24,32)(26,28)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,8,4)(2,10,12,6)(3,14,16,7)(9,11,13,15)(17,21,23,24)(18,19,26,22)(20,29,31,32)(25,27,28,30)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,48)(42,43);(1,6,8,10)(2,5,12,4)(3,15,16,11)(7,9,14,13)(17,22,23,19)(18,21,26,24)(20,30,31,27)(25,29,28,32)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,43,48);(1,7,8,14)(2,11,12,15)(3,4,16,5)(6,9,10,13)(17,20)(18,25)(19,27)(21,29)(22,30)(23,31)(24,32)(26,28)(33,37,38,36)(34,41,42,40)(35,45,46,44)(39,48,43,47)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8050349);