#8889441

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	#57752	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=g,e^2=i,f^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, 8796637);