#321774

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
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	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

751 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=egi,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi

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, 228970);