#464950

order	512
nilpotency class	3
isoclinism	isoclinic class of #453996
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1219 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,h^d=hi

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, 372146);