#8334397

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8241593);