#397403

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
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

1084 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,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,f^c=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304599);