#10451617

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3897 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)(3,11,14,8)(4,5,10,9)(12,16,15,13)(17,18)(19,24)(20,26)(21,31)(22,32)(23,28)(25,30)(27,29)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,47)(44,48);(2,7)(4,10)(8,11)(12,15)(17,19,23,31)(18,24,28,21)(20,22,30,29)(25,27,26,32);(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,32)(26,28)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,3)(2,8)(4,15)(5,16)(6,14)(7,11)(9,13)(10,12)(17,20)(18,26)(19,22)(21,27)(23,30)(24,32)(25,28)(29,31)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,4,6,10)(2,9,7,5)(3,12,14,15)(8,16,11,13)(17,21,23,24)(18,19,28,31)(20,27,30,32)(22,25,29,26)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,5,6,9)(2,10,7,4)(3,13,14,16)(8,15,11,12)(17,22)(18,27)(19,30)(20,31)(21,25)(23,29)(24,26)(28,32)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10358813);