#321728

order	512
nilpotency class	3
isoclinism	isoclinic class of #278235
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

765 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=eg,d^2=ei,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228924);