#340404

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

701 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=dei,e^b=eh,d^c=dh,f^c=fi,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247600);