#8436797

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=g,e^2=gh,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, 8343993);