#8762094

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	#51815	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4129 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,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, 8669290);