#10128024

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

4253 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=gi,e^2=g,f^2=g,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, 10035220);