#8923666

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	#59795	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8830862);