#8749948

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657144);