#9017361

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	#53841	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924557);