#8496878

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3906 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8404074);