#464889

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 372085);