#6825423

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#31021	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1513 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,23,30,36)(20,24,32,38)(21,25,28,34)(27,33,40,44)(29,35,42,46)(31,37,41,45)(39,43,47,48);(2,5)(3,7)(19,27)(20,31)(21,28)(23,33)(24,37)(25,34)(29,42)(30,40)(32,41)(35,46)(36,44)(38,45);(10,13)(14,16)(17,19,28,40,22,30,21,27)(18,23,34,44,26,36,25,33)(20,29,41,47,32,42,31,39)(24,35,45,48,38,46,37,43);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(20,32)(24,38)(29,42)(31,41)(35,46)(37,45)(39,47)(43,48);(1,3,4,7)(2,6,5,8)(17,20,21,31,22,32,28,41)(18,24,25,37,26,38,34,45)(19,29,27,39,30,42,40,47)(23,35,33,43,36,46,44,48)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6732619);