#6552537

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=f,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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, 6459733);