#321383

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

910 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)(5,6,8,10,7,9,11,12)(13,14,16,22,18,21,17,23)(15,19,24,28,26,27,25,20);(1,2,4,3)(6,9)(10,12)(13,15,16,24,18,26,17,25)(14,19,22,28,21,27,23,20);(5,6)(7,9)(8,12)(10,11)(13,15,18,26)(14,20,21,28)(16,25,17,24)(19,22,27,23);(1,3,4,2)(14,21)(19,27)(20,28)(22,23)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=i,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,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, 228579);