#8630835

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	#54485	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,23,25)(19,24,29,31)(20,26,27,21)(22,28,32,30)(33,34,39,38)(35,42,40,36)(37,45,46,41)(43,44,47,48);(2,7)(3,11)(4,14)(13,15)(17,19)(18,24)(20,30)(21,32)(22,26)(23,29)(25,31)(27,28)(33,35,39,40)(34,36,38,42)(37,43,46,47)(41,44,45,48);(2,7)(4,14)(8,10)(12,16)(18,25)(19,29)(21,26)(28,30)(33,36,39,42)(34,40,38,35)(37,44,46,48)(41,47,45,43);(1,3)(2,8)(4,12)(5,15)(6,11)(7,10)(9,13)(14,16)(17,20,23,27)(18,26,25,21)(19,28,29,30)(22,24,32,31)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,21,23,26)(18,27,25,20)(19,22,29,32)(24,30,31,28)(33,37,39,46)(34,41,38,45)(35,43,40,47)(36,44,42,48);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,22)(18,28)(19,26)(20,31)(21,29)(23,32)(24,27)(25,30)(33,38,39,34)(35,42,40,36)(37,45,46,41)(43,48,47,44)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8538031);