#8096464

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003660);