#8430490

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4149 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,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, 8337686);