#8622293

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 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)(3,8)(4,9)(5,7)(6,10)(11,14)(12,16)(13,15)(17,18,23,26)(19,21,24,29)(20,22,25,31)(27,28,30,32)(33,34,38,39)(35,42,37,40)(36,44,45,41)(43,47,46,48);(1,3,7,10)(2,8,5,6)(4,12,15,11)(9,16,13,14)(17,19,23,24)(18,21,26,29)(20,27,25,30)(22,28,31,32)(34,39)(36,45)(40,42)(43,46);(2,5)(3,10)(9,13)(11,12)(19,24)(21,29)(27,30)(28,32)(33,35,38,37)(34,40,39,42)(36,46,45,43)(41,48,44,47);(1,4,7,15)(2,9,5,13)(3,11,10,12)(6,16,8,14)(17,20)(18,22)(19,27)(21,28)(23,25)(24,30)(26,31)(29,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(1,5,7,2)(3,8,10,6)(4,13,15,9)(11,14,12,16)(17,21,23,29)(18,24,26,19)(20,28,25,32)(22,30,31,27)(33,36,38,45)(34,41,39,44)(35,43,37,46)(40,47,42,48);(1,6,7,8)(2,10,5,3)(4,14,15,16)(9,11,13,12)(17,22)(18,25)(19,28)(20,26)(21,30)(23,31)(24,32)(27,29)(33,37,38,35)(34,40,39,42)(36,46,45,43)(41,47,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8529489);