#8993754

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	#44136	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3927 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)(3,5)(4,9)(6,16)(7,10)(8,14)(11,13)(12,15)(17,18,22,21)(19,23,28,26)(20,24,29,30)(25,31,27,32)(33,34)(35,41)(36,42)(37,40)(38,39)(43,47)(44,48)(45,46);(1,3,7,6)(2,5,10,16)(4,12,14,11)(8,13,9,15)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(34,39)(35,42)(37,45)(44,47);(1,4,7,14)(2,8,10,9)(3,11,6,12)(5,15,16,13)(17,20,22,29)(18,24,21,30)(19,25,28,27)(23,31,26,32)(33,35)(34,36)(37,43)(38,42)(39,41)(40,44)(45,48)(46,47);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,10)(13,16)(17,21,22,18)(19,26,28,23)(20,24,29,30)(25,31,27,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,5)(2,3)(4,13)(6,10)(7,16)(8,12)(9,11)(14,15)(17,20)(18,24)(19,27)(21,30)(22,29)(23,32)(25,28)(26,31)(33,37,38,45)(34,40,39,46)(35,43,42,48)(36,44,41,47);(1,6,7,3)(2,5,10,16)(4,11,14,12)(8,13,9,15)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=hi,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, 8900950);