#8811195

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	#48601	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,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, 8718391);