#8806871

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	#37154	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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)(3,5)(4,9)(6,7)(8,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,24)(22,26)(23,28)(25,30)(27,31)(29,32)(33,34,39,40)(35,43,44,41)(36,47,48,37)(38,45,46,42);(2,7)(4,12)(5,8)(11,16)(17,19,22,28)(18,23,26,21)(20,27,30,32)(24,29,25,31)(34,40)(35,44)(37,47)(42,45);(2,7)(3,10)(4,12)(13,15)(17,20,22,30)(18,24,26,25)(19,27,28,32)(21,31,23,29)(33,35,39,44)(34,41,40,43)(36,45,48,42)(37,46,47,38);(1,3,6,10)(2,8,7,5)(4,11,12,16)(9,15,14,13)(17,20)(18,25)(19,27)(21,29)(22,30)(23,31)(24,26)(28,32)(33,36)(34,37)(35,45)(38,43)(39,48)(40,47)(41,46)(42,44);(1,4,6,12)(2,9,7,14)(3,11,10,16)(5,13,8,15)(17,21)(18,19)(20,29)(22,23)(24,32)(25,27)(26,28)(30,31)(33,37)(34,36)(35,46)(38,44)(39,47)(40,48)(41,45)(42,43);(1,5,6,8)(2,10,7,3)(4,13,12,15)(9,16,14,11)(17,19)(18,21)(20,27)(22,28)(23,26)(24,31)(25,29)(30,32)(33,38,39,46)(34,42,40,45)(35,37,44,47)(36,43,48,41)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=hi,e^2=hi,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, 8714067);