#8901590

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

3839 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)(19,26)(20,25)(21,22)(23,24)(27,32)(28,29)(30,31)(33,34,39,36)(35,42,41,38)(37,40,45,44)(43,48,47,46);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(18,24)(21,27)(25,26)(28,30)(37,45)(40,44)(43,47)(46,48);(2,6)(7,13)(8,14)(15,16)(17,19,23,20)(18,25,24,26)(21,28,27,30)(22,31,32,29)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,48)(45,47);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,20,23,19)(18,26,24,25)(21,28,27,30)(22,31,32,29)(33,36,39,34)(35,42,41,38)(37,44,45,40)(43,48,47,46);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,21,23,27)(18,22,24,32)(19,28,20,30)(25,31,26,29)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,47)(42,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,22)(18,27)(19,29)(20,31)(21,24)(23,32)(25,28)(26,30)(33,38,39,42)(34,41,36,35)(37,46,45,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808786);