#8810145

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	#44438	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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)(18,23)(19,26)(25,29)(27,32)(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)(17,18,22,23)(19,24,26,20)(21,29,30,25)(27,28,32,31)(37,45)(40,44)(43,47)(46,48);(2,3)(4,11)(8,10)(12,16)(19,26)(20,24)(27,32)(28,31)(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)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,29)(30,32)(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)(17,20,22,24)(18,19,23,26)(21,28,30,31)(25,27,29,32)(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)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(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,d^2=h,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, 8717341);