#8445682

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

3884 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=g,e^2=i,f^2=g,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, 8352878);