#9020527

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	#52533	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8927723);