#8773011

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	#54199	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4083 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)(3,9)(4,7)(5,8)(6,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31)(33,34)(35,41)(36,42)(37,45)(38,39)(40,46)(43,44)(47,48);(1,2,5,8)(3,6,10,9)(4,7,12,14)(11,13,16,15)(18,23)(20,25)(21,30)(26,32)(34,39)(35,42)(40,45)(43,48);(3,10)(4,12)(6,9)(7,14)(17,19,22,27)(18,20,23,25)(21,26,30,32)(24,28,29,31)(33,35)(34,36)(37,43)(38,42)(39,41)(40,44)(45,47)(46,48);(1,3,5,10)(2,6,8,9)(4,11,12,16)(7,13,14,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,4,5,12)(2,7,8,14)(3,11,10,16)(6,13,9,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,34)(35,36)(37,40)(38,39)(41,42)(43,44)(45,46)(47,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(33,37,38,46)(34,40,39,45)(35,43,42,48)(36,44,41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=gh,e^2=gh,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, 8680207);