#429037

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

1052 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 336233);