#8718971

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	#55785	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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)(3,8)(4,5)(6,11)(7,9)(10,16)(12,15)(13,14)(17,18,23,28)(19,29,31,24)(20,22,27,25)(21,32,30,26)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(1,3,7,11)(2,8,9,6)(4,14,10,12)(5,13,16,15)(17,19)(18,24)(20,21)(22,26)(23,31)(25,32)(27,30)(28,29)(34,39)(35,42)(41,44)(43,48);(2,9)(3,11)(4,10)(13,15)(17,20,23,27)(18,25,28,22)(19,21,31,30)(24,32,29,26)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4,7,10)(2,5,9,16)(3,12,11,14)(6,13,8,15)(17,20)(18,25)(19,30)(21,31)(22,28)(23,27)(24,26)(29,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,5)(2,10)(3,13)(4,9)(6,14)(7,16)(8,12)(11,15)(17,21,23,30)(18,26,28,32)(19,20,31,27)(22,29,25,24)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,6,7,8)(2,11,9,3)(4,15,10,13)(5,12,16,14)(17,22,23,25)(18,27,28,20)(19,26,31,32)(21,24,30,29)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8626167);