#8695315

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	#38676	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,22,20)(19,25,27,23)(21,24,30,28)(26,32,29,31)(34,38)(35,41)(40,44)(42,43);(2,3)(7,9)(8,10)(14,15)(17,19,22,27)(18,23,20,25)(21,26,30,29)(24,31,28,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,48)(44,45);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,18,22,20)(19,23,27,25)(21,24,30,28)(26,31,29,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,48)(44,45);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,35,37,41)(34,39,38,46)(36,43,45,42)(40,48,44,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,46)(42,48)(43,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,36,37,45)(34,40,38,44)(35,42,41,43)(39,47,46,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=h,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, 8602511);