#9851419

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	#49957	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9758615);