#9726141

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 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)(3,5)(4,11)(6,14)(7,8)(9,16)(10,15)(12,13)(17,18)(19,28)(20,26)(21,27)(22,24)(23,25)(29,32)(30,31)(33,34)(35,41)(36,45)(37,39)(38,42)(40,43)(44,46)(47,48);(1,3,7,4)(2,5,8,11)(6,10,16,12)(9,13,14,15)(17,19,23,22)(18,24,25,28)(20,29,27,31)(21,32,26,30)(33,35,40,37)(34,41,43,39)(36,44,38,47)(42,48,45,46);(2,8)(3,4)(9,14)(10,12)(18,25)(20,27)(24,28)(29,31)(33,36,40,38)(34,42,43,45)(35,44,37,47)(39,46,41,48);(1,4,7,3)(2,5,8,11)(6,12,16,10)(9,13,14,15)(17,20,23,27)(18,26,25,21)(19,29,22,31)(24,30,28,32)(33,37,40,35)(34,39,43,41)(36,44,38,47)(42,48,45,46);(1,5)(2,3)(4,8)(6,15)(7,11)(9,12)(10,14)(13,16)(17,21,23,26)(18,27,25,20)(19,30,22,32)(24,29,28,31)(33,38,40,36)(34,42,43,45)(35,44,37,47)(39,48,41,46);(1,6,7,16)(2,9,8,14)(3,10,4,12)(5,13,11,15)(17,22,23,19)(18,28,25,24)(20,31,27,29)(21,30,26,32)(33,39)(34,35)(36,46)(37,43)(38,48)(40,41)(42,44)(45,47)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=h,e^2=hi,f^2=gi,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, 9633337);