#10158478

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	#50087	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 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)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,24,26)(19,27,23,25)(20,22,21,31)(28,32,29,30)(33,34,38,41)(35,40,44,39)(36,37,47,45)(42,48,46,43);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19,24,23)(18,25,26,27)(20,28,21,29)(22,30,31,32)(33,34,38,41)(35,40,44,39)(36,45,47,37)(42,43,46,48);(2,7)(6,13)(8,11)(14,15)(17,20,24,21)(18,22,26,31)(19,28,23,29)(25,30,27,32)(33,35,38,44)(34,39,41,40)(36,42,47,46)(37,43,45,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,21,24,20)(18,22,26,31)(19,29,23,28)(25,30,27,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,41)(42,47)(45,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,22)(18,21)(19,30)(20,26)(23,32)(24,31)(25,29)(27,28)(33,36,38,47)(34,37,41,45)(35,42,44,46)(39,43,40,48);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,23,24,19)(18,25,26,27)(20,29,21,28)(22,30,31,32)(33,37)(34,36)(35,43)(38,45)(39,42)(40,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=g,e^2=i,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, 10065674);