#10195597

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	#57795	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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)(3,6)(4,12)(5,8)(7,14)(9,11)(10,16)(13,15)(17,18,24,22)(19,27,23,25)(20,26,21,30)(28,32,29,31)(33,34,39,43)(35,40,45,38)(36,41,44,48)(37,42,46,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,16)(12,14)(13,15)(17,19,24,23)(18,25,22,27)(20,28,21,29)(26,31,30,32)(33,35)(34,40)(36,46)(37,44)(38,43)(39,45)(41,47)(42,48);(1,2,5,8)(3,6,11,9)(4,7,14,12)(10,15,13,16)(17,20,24,21)(18,26,22,30)(19,28,23,29)(25,31,27,32)(33,36,39,44)(34,41,43,48)(35,37,45,46)(38,47,40,42);(1,2)(3,9)(4,7)(5,8)(6,11)(10,16)(12,14)(13,15)(17,21,24,20)(18,26,22,30)(19,29,23,28)(25,31,27,32)(33,35)(34,40)(36,37)(38,43)(39,45)(41,42)(44,46)(47,48);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,16,12,15)(17,22,24,18)(19,27,23,25)(20,30,21,26)(28,32,29,31)(33,37)(34,42)(35,44)(36,45)(38,41)(39,46)(40,48)(43,47);(1,4)(2,7)(3,10)(5,14)(6,15)(8,12)(9,16)(11,13)(17,23,24,19)(18,25,22,27)(20,29,21,28)(26,31,30,32)(33,38,39,40)(34,35,43,45)(36,47,44,42)(37,48,46,41)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,d^2=g,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, 10102793);