#8621190

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	#56286	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8528386);