#8813985

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	#50059	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8721181);