#6574574

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2047 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481770);