#8943816

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	#53738	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8851012);