#8227324

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4070 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:

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

Polycyclic presentation:

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

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, 8134520);