#9717783

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	#36795	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9624979);