#6830908

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#42887	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1565 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=f,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6738104);