#10188263

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	#39282	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4129 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)(3,12)(4,5)(6,16)(7,9)(8,13)(10,15)(11,14)(17,18,23,25)(19,20,28,24)(21,26,32,30)(22,29,31,27)(33,34,40,38)(35,44,46,41)(36,42,37,47)(39,43,45,48);(1,3,7,13)(2,8,9,12)(4,14,15,6)(5,16,10,11)(17,19,23,28)(18,24,25,20)(21,31,32,22)(26,29,30,27)(33,35)(34,41)(36,39)(37,45)(38,44)(40,46)(42,48)(43,47);(2,9)(3,13)(4,15)(11,16)(18,25)(20,24)(26,30)(27,29)(33,36,40,37)(34,42,38,47)(35,39,46,45)(41,48,44,43);(1,4)(2,10)(3,14)(5,9)(6,13)(7,15)(8,11)(12,16)(17,20)(18,19)(21,27)(22,30)(23,24)(25,28)(26,31)(29,32)(33,37,40,36)(34,42,38,47)(35,45,46,39)(41,48,44,43);(1,5,7,10)(2,4,9,15)(3,11,13,16)(6,12,14,8)(17,21,23,32)(18,26,25,30)(19,22,28,31)(20,29,24,27)(33,38,40,34)(35,44,46,41)(36,47,37,42)(39,43,45,48);(1,6,7,14)(2,11,9,16)(3,4,13,15)(5,12,10,8)(17,22)(18,27)(19,21)(20,30)(23,31)(24,26)(25,29)(28,32)(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=gh,d^2=gh,e^2=g,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, 10095459);