#8806827

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

3928 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)(3,9)(4,7)(5,8)(6,10)(11,15)(12,14)(13,16)(18,23)(19,25)(21,29)(27,32)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,2,5,8)(3,6,10,9)(4,7,12,14)(11,13,16,15)(17,18,22,23)(19,20,25,28)(21,30,29,24)(26,32,31,27)(35,41)(37,45)(39,42)(40,44);(3,10)(4,12)(6,9)(7,14)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,3,5,10)(2,6,8,9)(4,11,12,16)(7,13,14,15)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(1,4,5,12)(2,7,8,14)(3,11,10,16)(6,13,9,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,36,38,34)(35,39,41,42)(37,44,45,40)(43,46,47,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,27,28,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714023);