#10195730

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	#50005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4074 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:

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

Polycyclic presentation:

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

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, 10102926);