#10202023

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

4071 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=ghi,e^2=gh,f^2=hi,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, 10109219);