#6380669

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
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

2002 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)(13,14,20,17,19,24,18,23)(15,25,27,21,29,36,28,37)(16,30,32,38,34,22,33,31)(26,40,43,44,42,39,41,35);(5,7,9,8)(6,10,11,12)(13,15)(14,21)(16,26)(17,25)(18,27)(19,29)(20,28)(22,35)(23,36)(24,37)(30,44)(31,39)(32,41)(33,43)(34,42)(38,40);(5,8,9,7)(6,10,11,12)(13,16,19,34)(14,22,24,30)(15,26,29,42)(17,31,23,38)(18,33,20,32)(21,35,37,44)(25,39,36,40)(27,43,28,41);(13,17,20,24,19,23,18,14)(15,21,27,36,29,37,28,25)(16,31,32,30,34,38,33,22)(26,35,43,40,42,44,41,39);(13,18,19,20)(14,23,24,17)(15,27,29,28)(16,32,34,33)(21,36,37,25)(22,31,30,38)(26,41,42,43)(35,39,44,40)

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^a=di,d^b=df,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287865);