#10188268

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	#58185	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18,22,25)(19,23,29,32)(20,24,30,21)(26,28,27,31)(33,34)(35,44)(36,46)(37,41)(38,45)(39,40)(42,47)(43,48);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,32)(26,30)(34,40)(35,37)(42,46)(43,45);(17,20)(18,24)(19,26)(21,25)(22,30)(23,28)(27,29)(31,32)(33,35,39,37)(34,41,40,44)(36,43,47,45)(38,42,48,46);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,20)(18,24)(19,27)(21,25)(22,30)(23,31)(26,29)(28,32)(33,36,39,47)(34,42,40,46)(35,45,37,43)(38,44,48,41);(1,5,6,3)(2,7,9,10)(4,11,13,14)(8,16,12,15)(17,19)(18,23)(20,26)(21,31)(22,29)(24,28)(25,32)(27,30)(33,37,39,35)(34,41,40,44)(36,43,47,45)(38,46,48,42);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,13)(14,16)(17,21,22,24)(18,20,25,30)(19,28,29,31)(23,27,32,26)(33,38)(34,43)(35,42)(36,44)(37,46)(39,48)(40,45)(41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=h,e^2=g,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, 10095464);