#288794

order	512
nilpotency class	3
isoclinism	isoclinic class of #288787
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	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

718 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=ef,d^2=eh,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=dh,f^c=fi,g^c=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, 195990);