#8813981

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8721177);