#10195845

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	#56728	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=gh,e^2=gh,f^2=hi,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, 10103041);