#6380664

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=fi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287860);