#8533513

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	#57569	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,16)(14,15)(17,18)(19,27)(20,29)(21,23)(22,26)(24,28)(25,31)(30,32)(33,34,39,42)(35,40,45,36)(37,38,41,47)(43,44,48,46);(1,3)(2,4)(5,11)(6,10)(7,8)(9,12)(13,15)(14,16)(17,19,23,28)(18,24,21,27)(20,30,31,22)(25,32,29,26)(33,35,39,45)(34,40,42,36)(37,43,41,48)(38,44,47,46);(2,7)(3,10)(9,15)(11,16)(18,21)(24,27)(25,29)(26,32)(33,36,39,40)(34,35,42,45)(37,46,41,44)(38,43,47,48);(1,4,6,8)(2,3,7,10)(5,13,14,12)(9,16,15,11)(17,20,23,31)(18,25,21,29)(19,22,28,30)(24,26,27,32)(33,37)(34,38)(35,43)(36,46)(39,41)(40,44)(42,47)(45,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,13)(12,15)(14,16)(17,21,23,18)(19,27,28,24)(20,29,31,25)(22,32,30,26)(33,36)(34,35)(37,46)(38,43)(39,40)(41,44)(42,45)(47,48);(1,5,6,14)(2,9,7,15)(3,11,10,16)(4,12,8,13)(17,22)(18,26)(19,20)(21,32)(23,30)(24,25)(27,29)(28,31)(33,38)(34,41)(35,44)(36,43)(37,42)(39,47)(40,48)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8440709);