#428647

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1075 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)(5,6,9,8)(7,11,12,10)(13,14,19,23)(15,21,25,28)(16,26,22,18)(17,27,24,20);(5,7)(6,10)(8,11)(9,12)(13,15,19,25)(14,20,23,27)(16,21,22,28)(17,26,24,18);(5,8,9,6)(7,11,12,10)(13,16,18,14,19,22,26,23)(15,21,24,27,25,28,17,20);(2,4)(7,12)(10,11)(13,17,18,15,19,24,26,25)(14,21,22,27,23,28,16,20)

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=e,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,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, 335843);