#8157248

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	#52586	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4141 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8064444);