#8839232

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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)(3,7)(4,8)(5,14)(6,10)(9,15)(11,12)(13,16)(17,18,23,26)(19,24,21,20)(22,30,31,25)(27,28,29,32)(34,39)(35,37)(41,44)(43,45);(1,3)(2,7)(4,5)(6,11)(8,14)(9,16)(10,12)(13,15)(17,18,23,26)(19,24,21,20)(22,30,31,25)(27,28,29,32)(33,34,38,39)(35,40,37,42)(36,44,46,41)(43,48,45,47);(3,11)(4,13)(7,12)(8,16)(17,19,23,21)(18,20,26,24)(22,27,31,29)(25,28,30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4,6,13)(2,8,10,16)(3,5,11,15)(7,14,12,9)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,30)(31,32)(33,35,38,37)(34,40,39,42)(36,43,46,45)(41,47,44,48);(1,5)(2,9)(3,4)(6,15)(7,16)(8,12)(10,14)(11,13)(17,21,23,19)(18,24,26,20)(22,29,31,27)(25,32,30,28)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,2)(3,12)(4,8)(5,9)(6,10)(7,11)(13,16)(14,15)(17,22,23,31)(18,25,26,30)(19,27,21,29)(20,28,24,32)(33,37,38,35)(34,42,39,40)(36,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8746428);