#8750856

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	#36694	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3987 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(18,20)(23,25)(24,28)(31,32)(34,38)(36,43)(39,44)(42,47);(1,3,7,4)(2,8,11,14)(5,12,10,15)(6,13,9,16)(17,18,22,20)(19,25,27,23)(21,24,30,28)(26,32,29,31)(35,41)(39,44)(42,47)(46,48);(3,4)(8,14)(12,15)(13,16)(17,18,22,20)(19,25,27,23)(21,24,30,28)(26,32,29,31)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4,7,3)(2,8,11,14)(5,15,10,12)(6,13,9,16)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,5)(2,9)(3,12)(4,15)(6,11)(7,10)(8,16)(13,14)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48);(1,6,7,9)(2,10,11,5)(3,13,4,16)(8,15,14,12)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658052);