#351864

order	512
nilpotency class	3
isoclinism	isoclinic class of #347758
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

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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=f,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ci,g^d=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, 259060);