#8898858

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3942 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,15)(13,14)(17,18,22,25)(19,23,29,26)(20,21,24,30)(27,28,31,32)(33,34,37,40)(35,38,42,46)(36,43,44,39)(41,47,48,45);(1,3,7,4)(2,8,6,9)(5,12,14,11)(10,16,13,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,6)(8,9)(10,13)(15,16)(17,18,22,25)(19,26,29,23)(20,21,24,30)(27,32,31,28)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,7,3)(2,9,6,8)(5,11,14,12)(10,15,13,16)(17,20)(18,21)(19,27)(22,24)(23,28)(25,30)(26,32)(29,31);(1,5,7,14)(2,10,6,13)(3,11,4,12)(8,15,9,16)(17,21)(18,24)(19,28)(20,25)(22,30)(23,31)(26,27)(29,32)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=g,e^2=g,f^2=h,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, 8806054);