#9717468

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=hi,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, 9624664);