#407893

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

993 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=hi,c^2=e,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^c=gi

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, 315089);