#8436211

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	#52425	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,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, 8343407);