#8841723

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	#45680	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=ghi,f^2=g,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, 8748919);