#281820

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24150	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

895 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)(13,14,17,27,18,23,19,22)(15,21,26,28,24,20,16,25);(2,4)(5,6)(7,9)(8,11)(10,12)(13,15,19,16,18,24,17,26)(14,20,22,28,23,21,27,25);(6,9)(7,12)(10,11)(13,16)(14,21)(15,19)(17,24)(18,26)(20,23)(22,28)(25,27);(2,4)(13,17,18,19)(14,22,23,27)(15,16,24,26)(20,25,21,28)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 189016);