#8417045

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	#52872	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324241);