#10168472

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	#53934	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=i,e^2=ghi,f^2=gh,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, 10075668);