#9712874

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

3867 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9620070);