#10143527

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	#53535	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)(3,8)(4,10)(5,11)(6,16)(7,9)(12,13)(14,15)(17,18,22,20)(19,23,27,24)(21,28,29,25)(26,31,32,30)(33,34,39,43)(35,44,42,38)(36,47,46,40)(37,41,45,48);(1,3,7,6)(2,8,9,16)(4,12,11,14)(5,15,10,13)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35,39,42)(34,38,43,44)(36,45,46,37)(40,48,47,41);(2,9)(4,11)(8,16)(12,14)(17,19,22,27)(18,24,20,23)(21,26,29,32)(25,31,28,30)(33,36,39,46)(34,40,43,47)(35,45,42,37)(38,48,44,41);(1,4,7,11)(2,10,9,5)(3,12,6,14)(8,13,16,15)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,36)(34,40)(35,45)(37,42)(38,48)(39,46)(41,44)(43,47);(1,5,7,10)(2,11,9,4)(3,13,6,15)(8,12,16,14)(17,20,22,18)(19,23,27,24)(21,25,29,28)(26,31,32,30)(33,37,39,45)(34,41,43,48)(35,46,42,36)(38,47,44,40);(1,6,7,3)(2,8,9,16)(4,14,11,12)(5,15,10,13)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,38)(34,42)(35,43)(36,48)(37,40)(39,44)(41,46)(45,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=ghi,e^2=h,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, 10050723);