#9716853

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	#57196	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 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)(3,8)(4,13)(5,10)(6,14)(7,9)(11,16)(12,15)(17,18)(19,29)(20,23)(21,26)(22,27)(24,28)(25,31)(30,32)(33,34)(35,45)(36,37)(38,42)(39,47)(40,44)(41,48)(43,46);(1,3,7,11)(2,8,9,16)(4,5,14,15)(6,12,13,10)(17,19,24,31)(18,25,28,29)(20,30,21,22)(23,27,26,32)(33,35)(34,41)(36,46)(37,47)(38,39)(40,48)(42,43)(44,45);(2,9)(3,11)(4,14)(10,12)(17,20,24,21)(18,23,28,26)(19,30,31,22)(25,27,29,32)(33,36,40,38)(34,42,44,37)(35,46,48,39)(41,43,45,47);(1,4)(2,6)(3,5)(7,14)(8,12)(9,13)(10,16)(11,15)(17,21,24,20)(18,26,28,23)(19,22,31,30)(25,32,29,27)(33,37,40,42)(34,36,44,38)(35,47,48,43)(39,41,46,45);(1,5)(2,10)(3,4)(6,16)(7,15)(8,13)(9,12)(11,14)(17,22)(18,27)(19,20)(21,31)(23,29)(24,30)(25,26)(28,32)(33,38,40,36)(34,42,44,37)(35,46,48,39)(41,47,45,43);(1,6)(2,4)(3,12)(5,8)(7,13)(9,14)(10,11)(15,16)(17,23,24,26)(18,20,28,21)(19,27,31,32)(22,25,30,29)(33,39,40,46)(34,43,44,47)(35,38,48,36)(37,41,42,45)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9624049);