#268076

order	512
nilpotency class	3
isoclinism	isoclinic class of #268043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

977 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175272);