#8639903

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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)(3,13)(4,10)(5,7)(6,11)(8,12)(9,16)(14,15)(17,18)(19,23)(20,30)(21,22)(24,25)(26,31)(27,28)(29,32)(33,34,38,37)(35,39,41,43)(36,40,44,45)(42,46,47,48);(1,3,8,16)(2,9,12,13)(4,14,5,11)(6,7,15,10)(17,19,24,29)(18,23,25,32)(20,27,21,26)(22,31,30,28)(35,41)(39,43)(42,47)(46,48);(1,4,8,5)(2,10,12,7)(3,14,16,11)(6,13,15,9)(17,20,24,21)(18,22,25,30)(19,26,29,27)(23,28,32,31)(33,35,38,41)(34,39,37,43)(36,42,44,47)(40,46,45,48);(1,5,8,4)(2,10,12,7)(3,11,16,14)(6,13,15,9)(17,21,24,20)(18,22,25,30)(19,27,29,26)(23,28,32,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,6,8,15)(2,11,12,14)(3,10,16,7)(4,9,5,13)(17,22)(18,20)(19,28)(21,25)(23,26)(24,30)(27,32)(29,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,7,8,10)(2,5,12,4)(3,15,16,6)(9,11,13,14)(17,23,24,32)(18,19,25,29)(20,28,21,31)(22,26,30,27)(33,37,38,34)(35,43,41,39)(36,40,44,45)(42,46,47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8547099);