#9794301

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	#42057	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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:

(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,26)(20,28)(21,31)(22,24)(23,29)(25,30)(27,32)(33,34,40,37)(35,38,41,36)(39,45,47,42)(43,44,48,46);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,32)(30,31)(33,35)(34,38)(36,37)(39,43)(40,41)(42,46)(44,45)(47,48);(2,9)(3,4)(10,16)(11,12)(18,24)(20,30)(23,26)(27,31)(33,36,40,38)(34,41,37,35)(39,44,47,46)(42,48,45,43);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20,22,30)(18,25,24,28)(19,27,29,31)(21,23,32,26)(33,37,40,34)(35,36,41,38)(39,45,47,42)(43,44,48,46);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,19)(18,26)(20,27)(21,25)(22,29)(23,24)(28,32)(30,31)(33,38,40,36)(34,41,37,35)(39,44,47,46)(42,43,45,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,21)(18,27)(19,28)(20,23)(22,32)(24,31)(25,29)(26,30)(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=ghi,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 9701497);