#9719172

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	#51846	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(3,5)(4,13)(6,15)(7,8)(9,14)(10,12)(11,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,32)(33,34,40,43)(35,44,37,38)(36,45,46,41)(39,47,48,42);(1,3,7,11)(2,5,8,16)(4,12,14,6)(9,15,13,10)(17,19)(18,24)(20,22)(21,30)(23,31)(25,32)(26,27)(28,29)(33,35,40,37)(34,38,43,44)(36,42,46,47)(39,45,48,41);(2,8)(3,11)(4,14)(10,15)(17,20,23,21)(18,25,28,26)(19,22,31,30)(24,32,29,27)(33,36,40,46)(34,41,43,45)(35,42,37,47)(38,39,44,48);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,13)(10,16)(17,21,23,20)(18,26,28,25)(19,30,31,22)(24,27,29,32)(33,37,40,35)(34,38,43,44)(36,42,46,47)(39,41,48,45);(1,5)(2,3)(4,15)(6,13)(7,16)(8,11)(9,12)(10,14)(17,22,23,30)(18,27,28,32)(19,20,31,21)(24,26,29,25)(33,38)(34,35)(36,39)(37,43)(40,44)(41,42)(45,47)(46,48);(1,6,7,12)(2,10,8,15)(3,4,11,14)(5,9,16,13)(17,18)(19,24)(20,25)(21,26)(22,32)(23,28)(27,30)(29,31)(33,39)(34,42)(35,45)(36,44)(37,41)(38,46)(40,48)(43,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=ghi,f^2=i,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, 9626368);