#8662978

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4199 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570174);