#6388989

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1992 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=dfi,e^b=ei,g^c=gi

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, 6296185);