#8696167

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	#59738	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603363);