#8340085

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	#49647	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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)(3,8,6,10)(5,9,14,12)(11,15,13,16)(17,18,23,28)(19,26,21,24)(20,29,31,25)(22,27,32,30)(33,34,39,41)(35,42,45,38)(36,37,40,48)(43,46,47,44);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(17,19,23,21)(18,24,28,26)(20,30,31,27)(22,29,32,25)(33,35)(34,38)(36,43)(37,44)(39,45)(40,47)(41,42)(46,48);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,19,23,21)(18,24,28,26)(20,27,31,30)(22,25,32,29)(33,36,39,40)(34,37,41,48)(35,43,45,47)(38,44,42,46);(1,4,7,2)(3,10,6,8)(5,9,14,12)(11,15,13,16)(17,20)(18,25)(19,27)(21,30)(22,26)(23,31)(24,32)(28,29)(33,34)(35,38)(36,37)(39,41)(40,48)(42,45)(43,44)(46,47);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,21,23,19)(18,26,28,24)(20,27,31,30)(22,25,32,29)(33,37)(34,40)(35,44)(36,41)(38,47)(39,48)(42,43)(45,46);(1,6,7,3)(2,10,4,8)(5,13,14,11)(9,16,12,15)(17,22,23,32)(18,27,28,30)(19,29,21,25)(20,26,31,24)(33,38)(34,35)(36,46)(37,47)(39,42)(40,44)(41,45)(43,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=i,e^2=h,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, 8247281);