#8898866

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3927 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,e^2=g,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, 8806062);