#8772628

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679824);