#321723

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	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

793 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eghi,c^2=i,d^2=e,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, 228919);