#115823

order	512
nilpotency class	3
isoclinism	isoclinic class of #100027
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,f^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fi,f^c=fi,g^c=gh

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, 23019);