#8496587

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	#39785	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8403783);