#354350

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=gi,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=di,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261546);