#9716226

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3803 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)(3,5,9,15)(4,8,13,16)(10,14,11,12)(17,18,24,28)(19,29,32,25)(20,26,21,23)(22,27,30,31)(34,37)(39,41)(40,44)(46,47);(1,3)(2,5)(4,11)(6,9)(7,15)(8,12)(10,13)(14,16)(17,19,24,32)(18,25,28,29)(20,30,21,22)(23,31,26,27)(33,34,38,37)(35,39,43,41)(36,44,45,40)(42,47,48,46);(2,7)(3,9)(8,16)(10,11)(17,20,24,21)(18,26,28,23)(19,30,32,22)(25,27,29,31)(33,34,38,37)(35,41,43,39)(36,40,45,44)(42,47,48,46);(1,2)(3,5)(4,8)(6,7)(9,15)(10,14)(11,12)(13,16)(17,21,24,20)(18,23,28,26)(19,22,32,30)(25,31,29,27)(33,35,38,43)(34,39,37,41)(36,42,45,48)(40,46,44,47);(1,4,6,13)(2,8,7,16)(3,10,9,11)(5,14,15,12)(17,22)(18,27)(19,20)(21,32)(23,25)(24,30)(26,29)(28,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48);(1,5)(2,3)(4,12)(6,15)(7,9)(8,11)(10,16)(13,14)(17,23)(18,20)(19,31)(21,28)(22,29)(24,26)(25,30)(27,32)(33,37,38,34)(35,41,43,39)(36,40,45,44)(42,46,48,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=i,f^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, 9623422);