#10195837

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

4058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=gh,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, 10103033);