#8944485

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851681);