#8898518

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805714);