#10168458

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3844 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gi,e^2=ghi,f^2=h,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, 10075654);