#8734478

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	#40842	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,14)(13,15)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,38,41)(35,40,44,39)(36,45,46,37)(42,43,48,47);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,15,14,16)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25)(33,35,38,44)(34,39,41,40)(36,42,46,48)(37,43,45,47);(4,12)(8,14)(11,13)(15,16)(19,27)(20,28)(23,32)(24,29)(33,35,38,44)(34,39,41,40)(36,42,46,48)(37,43,45,47);(1,4,6,12)(2,8,9,14)(3,11,5,13)(7,15,10,16)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31)(33,35)(34,40)(36,42)(37,47)(38,44)(39,41)(43,45)(46,48);(1,5,6,3)(2,7,9,10)(4,13,12,11)(8,15,14,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29)(33,36,38,46)(34,37,41,45)(35,42,44,48)(39,43,40,47);(1,2)(3,7)(4,14)(5,10)(6,9)(8,12)(11,16)(13,15)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,37)(34,36)(35,43)(38,45)(39,42)(40,48)(41,46)(44,47)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8641674);