#8810087

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	#48247	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4073 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^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, 8717283);