#8581521

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	#57477	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=hi,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^c=egi,f^d=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, 8488717);