#444459

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1014 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:

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

Polycyclic presentation:

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

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, 351655);