#10529209

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	#58901	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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:

(1,2,6,7)(3,8,10,4)(5,9,14,15)(11,13,16,12)(17,18)(19,26)(20,25)(21,24)(22,23)(27,32)(28,31)(29,30)(33,34,38,41)(35,37,40,43)(36,39,46,48)(42,45,44,47);(2,7)(3,10)(9,15)(11,16)(17,19,23,21)(18,24,22,26)(20,28,30,27)(25,32,29,31)(33,35)(34,37)(36,44)(38,40)(39,47)(41,43)(42,46)(45,48);(17,19,23,21)(18,24,22,26)(20,27,30,28)(25,31,29,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,48)(43,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,13)(12,15)(14,16)(17,20)(18,25)(19,27)(21,28)(22,29)(23,30)(24,31)(26,32)(33,34)(35,43)(36,39)(37,40)(38,41)(42,47)(44,45)(46,48);(1,4,6,8)(2,3,7,10)(5,13,14,12)(9,16,15,11)(17,21,23,19)(18,24,22,26)(20,27,30,28)(25,32,29,31)(33,36)(34,39)(35,44)(37,47)(38,46)(40,42)(41,48)(43,45);(1,5,6,14)(2,9,7,15)(3,11,10,16)(4,12,8,13)(17,22,23,18)(19,26,21,24)(20,29,30,25)(27,32,28,31)(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=h,c^2=h,e^2=g,f^2=gi,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, 10436405);