#8550935

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	#37348	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=gh,e^2=gh,f^2=gi,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, 8458131);