#6447090

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(13,14)(15,20)(16,21)(17,28)(18,22)(19,23)(24,25)(26,27);(2,3)(5,7,9,8)(6,10,11,12)(13,14)(15,24)(16,27)(17,22)(18,28)(19,23)(20,25)(21,26);(1,2,4,3)(5,8,9,7)(6,10,11,12)(13,15,19,26)(14,20,23,27)(16,17,25,18)(21,28,24,22);(1,3,4,2)(13,16)(14,21)(15,17)(18,26)(19,25)(20,28)(22,27)(23,24);(13,17,19,18)(14,22,23,28)(15,25,26,16)(20,21,27,24)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354286);