#9794251

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	#59779	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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)(3,12)(4,6)(5,8)(7,9)(10,11)(13,16)(14,15)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34,40,37)(35,38,41,36)(39,45,47,42)(43,44,48,46);(1,3,7,5)(2,8,9,12)(4,13,11,15)(6,14,10,16)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(33,35)(34,38)(36,37)(39,43)(40,41)(42,46)(44,45)(47,48);(2,9)(4,11)(8,12)(13,15)(18,25)(19,30)(20,27)(28,32)(33,36,40,38)(34,41,37,35)(39,44,47,46)(42,48,45,43);(1,4,7,11)(2,10,9,6)(3,13,5,15)(8,16,12,14)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,37,40,34)(35,36,41,38)(39,45,47,42)(43,44,48,46);(1,5,7,3)(2,8,9,12)(4,15,11,13)(6,14,10,16)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,38,40,36)(34,41,37,35)(39,44,47,46)(42,43,45,48);(1,6,7,10)(2,11,9,4)(3,14,5,16)(8,15,12,13)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(33,39,40,47)(34,42,37,45)(35,43,41,48)(36,44,38,46)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9701447);