#8159072

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=g,e^2=gh,f^2=h,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, 8066268);