#10040725

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3814 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)(3,8)(4,12)(5,10)(6,14)(7,11)(9,15)(13,16)(17,18)(19,21)(20,26)(22,25)(23,24)(27,30)(28,31)(29,32)(33,34,38,41)(35,40,43,37)(36,39,45,48)(42,44,47,46);(1,3,7,6)(2,8,11,14)(4,13,15,10)(5,12,16,9)(18,24)(20,28)(21,25)(27,29)(33,35,38,43)(34,37,41,40)(36,42,45,47)(39,46,48,44);(1,4,7,15)(2,9,11,12)(3,10,6,13)(5,8,16,14)(17,19,23,22)(18,25,24,21)(20,29,28,27)(26,30,31,32)(33,36)(34,39)(35,42)(37,46)(38,45)(40,44)(41,48)(43,47);(1,4)(2,9)(3,10)(5,14)(6,13)(7,15)(8,16)(11,12)(17,20,23,28)(18,26,24,31)(19,27,22,29)(21,30,25,32)(33,35)(34,40)(36,42)(37,41)(38,43)(39,44)(45,47)(46,48);(1,5)(2,10)(3,9)(4,14)(6,12)(7,16)(8,15)(11,13)(17,21,23,25)(18,19,24,22)(20,30,28,32)(26,27,31,29)(33,37,38,40)(34,35,41,43)(36,44,45,46)(39,47,48,42);(1,6,7,3)(2,8,11,14)(4,13,15,10)(5,9,16,12)(17,22,23,19)(18,21,24,25)(20,29,28,27)(26,30,31,32)(33,36)(34,39)(35,42)(37,46)(38,45)(40,44)(41,48)(43,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=hi,e^2=g,f^2=h,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, 9947921);