#10168361

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	#56502	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3901 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)(4,5)(6,14)(7,9)(10,16)(11,13)(12,15)(17,18,24,22)(19,28,30,25)(20,26,21,31)(23,27,29,32)(33,34,39,44)(35,42,46,40)(36,47,43,38)(37,45,48,41);(1,3,7,4)(2,8,9,5)(6,11,16,12)(10,15,14,13)(17,19)(18,25)(20,23)(21,29)(22,28)(24,30)(26,32)(27,31)(33,35,39,46)(34,40,44,42)(36,37,43,48)(38,45,47,41);(2,9)(3,4)(10,14)(11,12)(17,20,24,21)(18,26,22,31)(19,23,30,29)(25,32,28,27)(33,36,39,43)(34,38,44,47)(35,37,46,48)(40,45,42,41);(1,4,7,3)(2,5,9,8)(6,12,16,11)(10,13,14,15)(17,21,24,20)(18,26,22,31)(19,29,30,23)(25,32,28,27)(33,37,39,48)(34,41,44,45)(35,43,46,36)(38,42,47,40);(1,5)(2,4)(3,9)(6,15)(7,8)(10,11)(12,14)(13,16)(17,22,24,18)(19,28,30,25)(20,31,21,26)(23,27,29,32)(33,35)(34,42)(36,37)(38,41)(39,46)(40,44)(43,48)(45,47);(1,6,7,16)(2,10,9,14)(3,11,4,12)(5,13,8,15)(17,23,24,29)(18,27,22,32)(19,20,30,21)(25,31,28,26)(33,38)(34,43)(35,45)(36,44)(37,42)(39,47)(40,48)(41,46)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gi,e^2=ghi,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, 10075557);