#8963642

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	#48580	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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)(3,9,4,8)(6,13,14,10)(11,15,12,16)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,18,22,20)(19,23,26,27)(21,24,29,28)(25,30,31,32)(36,44)(39,46)(42,48)(45,47);(2,5)(6,14)(8,9)(11,12)(18,20)(21,29)(23,27)(25,31)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,4,7,3)(2,9,5,8)(6,12,14,11)(10,16,13,15)(17,19,22,26)(18,23,20,27)(21,25,29,31)(24,30,28,32);(1,5,7,2)(3,8,4,9)(6,13,14,10)(11,15,12,16)(17,20,22,18)(19,23,26,27)(21,28,29,24)(25,30,31,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,6,7,14)(2,10,5,13)(3,11,4,12)(8,15,9,16)(17,21)(18,24)(19,25)(20,28)(22,29)(23,30)(26,31)(27,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=gi,e^2=gi,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, 8870838);