#9712855

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	#51926	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=gh,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, 9620051);