#8679881

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3809 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=gh,e^2=h,f^2=gi,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, 8587077);