#8227322

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	#41474	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,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, 8134518);