#8695221

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

3869 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=gh,e^2=h,f^2=h,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, 8602417);