#422597

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1027 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=e,d^2=ei,f^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=dei,f^c=fi,g^c=gi,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, 329793);