#109982

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

596 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240051
maximal normal subgroup of	#72240051
maximal quotient of	#72240054, #72240056

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17178);