#6557405

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fi,d^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6464601);