#10506273

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	#56909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10413469);