#8550993

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	#40229	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8458189);