#8436749

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3968 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)(3,13)(4,10)(5,7)(6,15)(8,12)(9,16)(11,14)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32)(33,34,39,38)(35,42,40,36)(37,45,47,41)(43,44,46,48);(1,3)(2,9)(4,6)(5,14)(7,15)(8,16)(10,11)(12,13)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25)(33,35)(34,36)(37,46)(38,42)(39,40)(41,48)(43,47)(44,45);(1,4,8,5)(2,10,12,7)(3,6,16,14)(9,11,13,15)(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32)(33,36,39,42)(34,40,38,35)(37,44,47,48)(41,46,45,43);(1,5,8,4)(2,10,12,7)(3,14,16,6)(9,11,13,15)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32)(33,37,39,47)(34,41,38,45)(35,43,40,46)(36,44,42,48);(1,6,8,14)(2,11,12,15)(3,4,16,5)(7,9,10,13)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30);(1,7,8,10)(2,5,12,4)(3,15,16,11)(6,9,14,13)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)(33,38,39,34)(35,42,40,36)(37,41,47,45)(43,44,46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=g,e^2=gh,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, 8343945);