#8922799

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	#58238	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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829995);