#8504909

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=i,e^2=g,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, 8412105);