#10521354

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

4194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=hi,e^2=i,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, 10428550);