#9720443

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	#54119	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4027 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)(3,11)(4,9)(5,15)(6,8)(7,12)(10,13)(14,16)(17,18,22,27)(19,23,30,24)(20,25,26,21)(28,29,31,32)(34,39)(35,42)(37,45)(44,47);(1,3,6,12)(2,7,8,11)(4,13,14,5)(9,15,16,10)(17,19)(18,23)(20,31)(21,29)(22,30)(24,27)(25,32)(26,28)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(2,8)(7,11)(9,16)(10,15)(17,19,22,30)(18,24,27,23)(20,28,26,31)(21,29,25,32)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,2)(3,7)(4,9)(5,10)(6,8)(11,12)(13,15)(14,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,27)(25,32)(26,31)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,4,6,14)(2,9,8,16)(3,5,12,13)(7,10,11,15)(17,20)(18,25)(19,28)(21,27)(22,26)(23,29)(24,32)(30,31)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,5)(2,10)(3,4)(6,13)(7,9)(8,15)(11,16)(12,14)(17,21,22,25)(18,26,27,20)(19,29,30,32)(23,28,24,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,e^2=hi,f^2=gh,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, 9627639);