#10201921

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3862 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=gi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10109117);