#8372231

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=h,f^2=gi,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, 8279427);