#10196205

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	#56610	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4074 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=h,e^2=g,f^2=i,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, 10103401);