#351820

order	512
nilpotency class	3
isoclinism	isoclinic class of #348358
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

877 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ch,d^c=dh,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259016);