#8898645

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	#40118	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4058 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)(3,10)(4,13)(5,9)(6,7)(8,12)(11,15)(14,16)(17,18)(19,24)(20,30)(21,25)(22,28)(23,27)(26,32)(29,31)(33,34,39,38)(35,41,45,40)(36,46,44,42)(37,43,48,47);(1,3)(2,8)(4,11)(5,14)(6,10)(7,12)(9,15)(13,16)(17,19,23,29)(18,24,27,31)(20,28,21,26)(22,25,32,30)(33,35,39,45)(34,40,38,41)(36,37,44,48)(42,43,46,47);(1,4,7,5)(2,9,6,13)(3,11,12,14)(8,15,10,16)(17,20,23,21)(18,25,27,30)(19,26,29,28)(22,31,32,24)(33,35,39,45)(34,41,38,40)(36,37,44,48)(42,47,46,43);(1,3)(2,8)(4,11)(5,14)(6,10)(7,12)(9,15)(13,16)(17,18)(19,24)(20,25)(21,30)(22,26)(23,27)(28,32)(29,31)(33,36)(34,42)(35,37)(38,46)(39,44)(40,43)(41,47)(45,48);(1,5,7,4)(2,9,6,13)(3,11,12,14)(8,16,10,15)(17,21,23,20)(18,25,27,30)(19,28,29,26)(22,31,32,24)(33,37)(34,43)(35,44)(36,45)(38,47)(39,48)(40,46)(41,42);(1,6,7,2)(3,10,12,8)(4,9,5,13)(11,15,14,16)(17,22)(18,26)(19,25)(20,24)(21,31)(23,32)(27,28)(29,30)(33,38,39,34)(35,41,45,40)(36,46,44,42)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,e^2=h,f^2=g,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, 8805841);