#9863086

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

3812 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770282);