#8463425

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370621);