#298254

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#28437	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1214 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)(5,6,9,11)(7,8,12,10)(13,14,17,20)(15,22,24,19)(16,26,21,18)(23,28,27,25);(1,2,4,3)(5,7,9,12)(6,10,11,8)(13,15)(14,19)(16,27)(17,24)(18,25)(20,22)(21,23)(26,28);(5,7,10,6,9,12,8,11)(13,16,18,14,17,21,26,20)(15,23,25,22,24,27,28,19);(1,3,4,2)(5,8,9,10)(6,7,11,12)(14,20)(15,24)(16,21)(25,28)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=f,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205450);