#8873804

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8781000);