#10521360

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4216 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=ghi,e^2=gi,f^2=g,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, 10428556);