#10506269

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	#45637	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4084 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10413465);