#10216493

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	#44078	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,e^2=ghi,f^2=ghi,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, 10123689);