#8533231

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	#54019	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,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, 8440427);