#8581755

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	#57964	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4221 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)(3,11)(4,14)(5,10)(6,8)(7,13)(9,15)(12,16)(18,23)(19,25)(24,30)(26,31)(33,34,38,41)(35,42,36,39)(37,40,45,48)(43,47,44,46);(1,3,6,13)(2,7,8,11)(4,12,15,10)(5,9,16,14)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(35,36)(39,42)(43,44)(46,47);(2,8)(4,15)(7,11)(10,12)(20,27)(24,30)(26,31)(28,32)(33,35,38,36)(34,39,41,42)(37,43,45,44)(40,46,48,47);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,36,38,35)(34,39,41,42)(37,43,45,44)(40,47,48,46);(1,4)(2,9)(3,12)(5,11)(6,15)(7,16)(8,14)(10,13)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,46)(45,48);(1,5,6,16)(2,10,8,12)(3,9,13,14)(4,11,15,7)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,48)(42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=i,e^2=ghi,f^2=hi,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, 8488951);