#8957590

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	#54926	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18,23,20)(19,24,26,22)(21,25,29,28)(27,31,32,30)(33,34)(35,36)(37,45)(38,41)(39,42)(40,47)(43,48)(44,46);(2,6)(7,9)(8,12)(14,15)(17,19,23,26)(18,24,20,22)(21,27,29,32)(25,31,28,30)(35,42)(36,39)(43,46)(44,48);(2,6)(4,13)(7,9)(10,16)(19,26)(21,29)(22,24)(25,28)(33,35,38,42)(34,39,41,36)(37,46,47,43)(40,44,45,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,23,18)(19,22,26,24)(21,25,29,28)(27,31,32,30)(33,34)(35,39)(36,42)(37,40)(38,41)(43,48)(44,46)(45,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,36,38,39)(34,35,41,42)(37,44,47,48)(40,43,45,46);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,22)(18,19)(20,26)(21,30)(23,24)(25,27)(28,32)(29,31)(33,37,38,47)(34,40,41,45)(35,43,42,46)(36,44,39,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8864786);