#8810162

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=h,e^2=h,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, 8717358);