#8810157

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	#56532	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,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, 8717353);