#6375767

order	512
nilpotency class	3
isoclinism	isoclinic class of #60771
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1916 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)(6,9)(7,11)(13,14)(15,22)(16,18)(17,19)(20,23)(21,25)(24,27)(26,28);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(14,18)(19,21)(20,22)(26,27);(6,9)(10,12)(14,19)(17,25)(18,21)(20,26)(22,27)(24,28);(5,7,8,11)(6,10,9,12)(13,15,16,23)(14,20,18,22)(17,24,25,28)(19,26,21,27);(15,23)(20,22)(24,28)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6282963);