#8139644

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046840);