#281666

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 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)(13,14,16,20,18,22,17,21)(15,23,26,28,27,24,25,19);(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,16,20,18,22,17,21)(15,24,26,19,27,23,25,28);(6,10)(7,8)(11,12)(13,15,18,27)(14,19,22,28)(16,25,17,26)(20,24,21,23);(2,4)(5,7,9,8)(6,11,12,10)(13,16,18,17)(14,20,22,21)(15,25,27,26)(19,24,28,23)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188862);