#8174309

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8081505);