#6458500

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	#34633	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365696);