#8810191

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	#58892	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=h,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, 8717387);