#8602623

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

3908 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=gh,e^2=g,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, 8509819);