#10106956

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	#43210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3927 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,15)(14,16)(17,18,22,19)(20,23,28,25)(21,26,30,24)(27,32,29,31)(33,34)(35,44)(36,42)(37,38)(39,41)(40,43)(45,47)(46,48);(2,7)(4,14)(8,11)(10,13)(17,18,22,19)(20,25,28,23)(21,24,30,26)(27,32,29,31)(33,35,40,39)(34,41,43,44)(36,45,37,46)(38,47,42,48);(2,7)(3,12)(9,16)(10,13)(18,19)(23,25)(24,26)(31,32)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,48,44,47);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,14)(15,16)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,37,40,36)(34,42,43,38)(35,46,39,45)(41,48,44,47);(1,4,6,14)(2,9,7,16)(3,13,12,10)(5,8,15,11)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30)(33,38,40,42)(34,37,43,36)(35,47,39,48)(41,46,44,45);(1,5,6,15)(2,10,7,13)(3,9,12,16)(4,11,14,8)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,47,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=gh,e^2=g,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, 10014152);