#8270531

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	#56413	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4300 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=gi,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8177727);