#9815364

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	#55424	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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)(3,11,13,8)(4,14,15,6)(5,9,16,12)(17,18)(19,24)(20,21)(22,31)(23,27)(25,30)(26,28)(29,32)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(1,3,7,13)(2,8,10,11)(4,12,15,9)(5,14,16,6)(17,19,23,29)(18,24,27,32)(20,28,30,22)(21,26,25,31)(35,41)(39,42)(43,45)(47,48);(17,20,23,30)(18,25,27,21)(19,22,29,28)(24,26,32,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,4)(2,6)(3,12)(5,8)(7,15)(9,13)(10,14)(11,16)(17,20)(18,21)(19,28)(22,29)(23,30)(24,26)(25,27)(31,32)(33,36,38,34)(35,42,41,39)(37,44,46,40)(43,48,45,47);(1,5,7,16)(2,9,10,12)(3,6,13,14)(4,8,15,11)(17,21)(18,20)(19,26)(22,32)(23,25)(24,28)(27,30)(29,31)(33,35)(34,39)(36,42)(37,45)(38,41)(40,48)(43,46)(44,47);(1,6)(2,4)(3,5)(7,14)(8,12)(9,11)(10,15)(13,16)(17,22,23,28)(18,26,27,31)(19,20,29,30)(21,24,25,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=hi,e^2=g,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, 9722560);