#10069049

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3973 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9976245);