#439572

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1281 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)(5,6,9,11)(7,10,8,12)(13,14,18,22,19,23,24,17)(15,16,25,27,26,28,21,20);(5,7,9,8)(6,10,11,12)(13,15,24,25,19,26,18,21)(14,20,17,16,23,27,22,28);(5,8,9,7)(6,10,11,12)(13,16,18,27,19,28,24,20)(14,21,22,15,23,25,17,26);(2,4)(13,17,18,14,19,22,24,23)(15,20,25,16,26,27,21,28)

Polycyclic presentation:

2^9|a^2=efi,b^2=efg,c^2=efi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi

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, 346768);