#10127728

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 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)(17,18,22,24)(19,20,27,23)(21,25,31,32)(26,28,29,30)(33,34,39,36)(35,40,43,44)(37,41,47,46)(38,45,48,42);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,35)(34,40)(36,44)(37,42)(38,41)(39,43)(45,47)(46,48);(2,7)(6,13)(8,11)(14,15)(18,24)(20,23)(25,32)(28,30)(34,36)(35,43)(41,46)(42,45);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,22,23)(18,19,24,27)(21,30,31,28)(25,29,32,26)(33,36,39,34)(35,44,43,40)(37,41,47,46)(38,45,48,42);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21,22,31)(18,25,24,32)(19,26,27,29)(20,28,23,30)(33,37)(34,41)(35,45)(36,46)(38,44)(39,47)(40,48)(42,43);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,38)(34,42)(35,41)(36,45)(37,40)(39,48)(43,46)(44,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=gi,e^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, 10034924);