#8436219

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	#37496	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4070 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:

(1,2)(3,13)(4,10)(5,7)(6,16)(8,12)(9,15)(11,14)(17,18,24,27)(19,28,23,25)(20,22,21,26)(29,31,30,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,3,8,15)(2,9,12,13)(4,14,5,6)(7,11,10,16)(17,19,24,23)(18,25,27,28)(20,30,21,29)(22,31,26,32)(34,38)(39,41)(40,45)(46,47);(1,4,8,5)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,31,28,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,5,8,4)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,21,24,20)(18,22,27,26)(19,30,23,29)(25,32,28,31)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,6)(2,11)(3,4)(5,15)(7,13)(8,14)(9,10)(12,16)(17,22)(18,20)(19,31)(21,27)(23,32)(24,26)(25,30)(28,29);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,13,14,9)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,32,26,31)(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,d^2=g,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, 8343415);