#10127733

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	#49909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^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, 10034929);