#8434117

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	#43202	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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)(3,11)(4,15)(5,16)(6,10)(7,14)(8,13)(9,12)(20,28)(24,31)(27,32)(29,30)(33,34,40,38)(35,45,46,41)(36,43,37,42)(39,44,47,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,9)(10,11)(15,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,35,40,46)(34,41,38,45)(36,39,37,47)(42,44,43,48);(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36,40,37)(34,42,38,43)(35,39,46,47)(41,44,45,48);(1,4,6,13)(2,8,10,15)(3,12,14,5)(7,9,11,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,37,40,36)(34,43,38,42)(35,39,46,47)(41,44,45,48);(1,5,6,12)(2,9,10,16)(3,13,14,4)(7,8,11,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,38,40,34)(35,45,46,41)(36,43,37,42)(39,48,47,44);(1,2)(3,7)(4,15)(5,9)(6,10)(8,13)(11,14)(12,16)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,39)(34,44)(35,37)(36,46)(38,48)(40,47)(41,43)(42,45)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=g,e^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, 8341313);