#8813961

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	#38325	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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)(17,18,22,27)(19,26,30,23)(20,29,31,24)(21,25,32,28);(1,3,6,11)(2,4,7,9)(5,14,15,10)(8,16,13,12)(17,19,22,30)(18,23,27,26)(20,28,31,25)(21,29,32,24)(33,34,38,40)(35,41,43,36)(37,45,46,39)(42,44,48,47);(2,7)(4,9)(8,13)(12,16)(20,31)(21,32)(24,29)(25,28)(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)(17,20,22,31)(18,24,27,29)(19,28,30,25)(21,26,32,23)(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)(17,21,22,32)(18,25,27,28)(19,29,30,24)(20,26,31,23)(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)(17,19)(18,26)(20,28)(21,29)(22,30)(23,27)(24,32)(25,31)(33,37,38,46)(34,39,40,45)(35,42,43,48)(36,44,41,47)

Polycyclic presentation:

2^9|a^2=g,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, 8721157);