#10168363

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	#48200	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30)(33,34,39,43)(35,36,44,40)(37,45,47,41)(38,42,46,48);(1,3,7,4)(2,8,9,5)(6,11,16,12)(10,15,14,13)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32)(33,35)(34,40)(36,43)(37,42)(38,45)(39,44)(41,46)(47,48);(2,9)(3,4)(10,14)(11,12)(17,19,21,27)(18,22,24,25)(20,26,29,28)(23,30,32,31)(33,36,39,40)(34,35,43,44)(37,46,47,38)(41,48,45,42);(1,4,7,3)(2,5,9,8)(6,12,16,11)(10,13,14,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,37,39,47)(34,41,43,45)(35,42,44,48)(36,38,40,46);(1,5)(2,4)(3,9)(6,15)(7,8)(10,11)(12,14)(13,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)(33,38)(34,42)(35,41)(36,37)(39,46)(40,47)(43,48)(44,45);(1,6,7,16)(2,10,9,14)(3,11,4,12)(5,13,8,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,35)(34,40)(36,43)(37,42)(38,45)(39,44)(41,46)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,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, 10075559);