#9815315

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	#55601	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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)(3,9)(4,5)(6,7)(8,12)(10,15)(11,16)(13,14)(17,18)(19,26)(20,28)(21,24)(22,23)(25,30)(27,31)(29,32)(33,34,39,44)(35,40,46,37)(36,42,48,41)(38,47,45,43);(2,7)(4,13)(8,9)(10,16)(17,19,23,21)(18,24,22,26)(20,27,30,29)(25,31,28,32)(33,35,39,46)(34,40,44,37)(36,45,48,38)(41,47,42,43);(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,36,39,48)(34,41,44,42)(35,38,46,45)(37,47,40,43);(1,3,6,12)(2,8,7,9)(4,10,13,16)(5,11,14,15)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,36)(34,42)(35,45)(37,47)(38,46)(39,48)(40,43)(41,44);(1,4,6,13)(2,5,7,14)(3,10,12,16)(8,11,9,15)(17,21,23,19)(18,24,22,26)(20,29,30,27)(25,31,28,32)(33,37)(34,35)(36,47)(38,41)(39,40)(42,45)(43,48)(44,46);(1,5)(2,4)(3,11)(6,14)(7,13)(8,10)(9,16)(12,15)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)(33,38,39,45)(34,43,44,47)(35,36,46,48)(37,42,40,41)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=ghi,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, 9722511);