#10166053

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,21)(19,27,29,24)(20,28,30,25)(22,26,32,31)(33,34,39,43)(35,36,44,40)(37,45,47,41)(38,42,46,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,24)(20,26)(21,27)(22,28)(23,29)(25,32)(30,31)(33,35)(34,40)(36,43)(37,42)(38,45)(39,44)(41,46)(47,48);(2,3)(7,9)(8,10)(14,15)(17,18,23,21)(19,24,29,27)(20,25,30,28)(22,26,32,31)(33,36,39,40)(34,35,43,44)(37,46,47,38)(41,48,45,42);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,25)(19,26)(21,28)(22,27)(23,30)(24,32)(29,31)(33,37,39,47)(34,41,43,45)(35,42,44,48)(36,38,40,46);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,18)(19,27,29,24)(20,25,30,28)(22,26,32,31)(33,38)(34,42)(35,41)(36,37)(39,46)(40,47)(43,48)(44,45);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22,23,32)(18,26,21,31)(19,28,29,25)(20,27,30,24)(33,35)(34,40)(36,43)(37,42)(38,45)(39,44)(41,46)(47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=h,e^2=ghi,f^2=gh,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, 10073249);