#8374219

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	#55096	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8281415);