#10143420

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=hi,e^2=h,f^2=gh,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, 10050616);