#323124

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

806 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eghi,c^2=h,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230320);