#6370560

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2126 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,e^2=g,h^2=i,b^a=bf,d^a=dfi,e^a=ei,c^b=cfi,e^c=ei,e^d=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277756);