#6457672

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

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, 6364868);