#110072

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

878 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=ci,d^b=dh,e^c=eh,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239509
maximal normal subgroup of	#72239509
maximal quotient of	#72239511

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17268);