#109775

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=di,e^b=eh,d^c=dh,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240832, #72257926, #72257929, …
maximal normal subgroup of	#72240832, #72257926, #72257929, …
maximal quotient of	#72240834, #72240835

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16971);