#10153134

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,d^2=h,e^2=gi,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, 10060330);