#9863094

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

3738 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=i,e^2=ghi,f^2=gh,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, 9770290);