#8810171

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

3835 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)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(4,12)(8,13)(9,15)(14,16)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30)(34,38)(36,44)(39,41)(42,48);(1,2,6,5)(3,7,11,10)(4,8,12,13)(9,14,15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3,6,11)(2,7,5,10)(4,9,12,15)(8,14,13,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4,6,12)(2,8,5,13)(3,9,11,15)(7,14,10,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47);(1,5,6,2)(3,10,11,7)(4,13,12,8)(9,16,15,14)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=gh,e^2=h,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, 8717367);