#8978387

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	#55314	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8885583);