#10529829

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3861 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=g,e^2=gh,f^2=hi,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, 10437025);