#8761971

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	#49681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 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)(3,12)(4,13)(5,11)(6,16)(7,9)(8,14)(10,15)(17,18,22,25)(19,23,28,24)(20,21,30,32)(26,27,29,31)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3,7,14)(2,8,9,12)(4,5,15,16)(6,10,11,13)(17,19)(18,23)(20,29)(21,31)(22,28)(24,25)(26,30)(27,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,9)(4,15)(5,16)(8,12)(17,19,22,28)(18,24,25,23)(20,29,30,26)(21,27,32,31)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,4)(2,10)(3,5)(6,12)(7,15)(8,11)(9,13)(14,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,25)(29,30)(31,32)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,5)(2,11)(3,4)(6,9)(7,16)(8,10)(12,13)(14,15)(17,20)(18,21)(19,26)(22,30)(23,27)(24,31)(25,32)(28,29)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,6,7,11)(2,5,9,16)(3,13,14,10)(4,12,15,8)(17,21)(18,20)(19,27)(22,32)(23,26)(24,29)(25,30)(28,31)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,f^2=g,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, 8669167);