#8436841

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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)(3,11)(4,10)(5,7)(6,9)(8,12)(13,15)(14,16)(17,18,23,27)(19,28,29,24)(20,25,31,30)(21,22,26,32)(33,34,39,38)(35,42,40,36)(37,45,47,41)(43,44,46,48);(1,3,8,6)(2,9,12,11)(4,13,5,14)(7,15,10,16)(19,29)(20,31)(24,28)(25,30)(33,35)(34,36)(37,46)(38,42)(39,40)(41,48)(43,47)(44,45);(1,4,8,5)(2,10,12,7)(3,13,6,14)(9,16,11,15)(17,19,23,29)(18,24,27,28)(20,26,31,21)(22,30,32,25)(33,36,39,42)(34,40,38,35)(37,44,47,48)(41,46,45,43);(1,5,8,4)(2,10,12,7)(3,14,6,13)(9,16,11,15)(17,20)(18,25)(19,26)(21,29)(22,24)(23,31)(27,30)(28,32)(33,37,39,47)(34,41,38,45)(35,43,40,46)(36,44,42,48);(1,6,8,3)(2,11,12,9)(4,14,5,13)(7,16,10,15)(17,21,23,26)(18,22,27,32)(19,20,29,31)(24,30,28,25);(1,7,8,10)(2,5,12,4)(3,15,6,16)(9,14,11,13)(17,22)(18,26)(19,30)(20,28)(21,27)(23,32)(24,31)(25,29)(33,38,39,34)(35,42,40,36)(37,41,47,45)(43,44,46,48)

Polycyclic presentation:

2^9|a^2=h,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, 8344037);