#9020514

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	#47377	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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)(19,27)(20,31)(21,26)(22,23)(24,30)(25,29)(28,32)(33,34,38,43)(35,41,44,39)(36,40,42,37)(45,46,47,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,15)(12,13)(14,16)(17,19,23,30)(18,24,22,27)(20,28,29,21)(25,32,31,26)(35,44)(39,41)(45,47)(46,48);(3,10)(4,12)(5,7)(8,16)(18,22)(20,29)(21,28)(24,27)(33,35,38,44)(34,39,43,41)(36,47,42,45)(37,48,40,46);(1,3)(2,7)(4,13)(5,9)(6,10)(8,14)(11,12)(15,16)(17,20)(18,25)(19,28)(21,30)(22,31)(23,29)(24,32)(26,27)(33,36,38,42)(34,40,43,37)(35,45,44,47)(39,48,41,46);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,21,23,28)(18,26,22,32)(19,29,30,20)(24,31,27,25)(33,35)(34,41)(36,47)(37,46)(38,44)(39,43)(40,48)(42,45);(1,5,6,7)(2,3,9,10)(4,14,12,15)(8,11,16,13)(17,22,23,18)(19,27,30,24)(20,31,29,25)(21,32,28,26)(33,37,38,40)(34,42,43,36)(35,46,44,48)(39,47,41,45)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=gh,e^2=g,f^2=gi,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, 8927710);