#322055

order	512
nilpotency class	3
isoclinism	isoclinic class of #288715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

836 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eghi,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^c=fi,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229251);