#8157391

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	#57185	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=h,e^2=h,f^2=g,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, 8064587);