#8695323

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	#41268	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4036 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=h,e^2=h,f^2=gh,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, 8602519);