#110177

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×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	#386
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

770 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)(6,7,11,12)(13,14,17,22,18,19,16,21)(15,23,26,20,24,28,25,27);(5,6,9,7,10,11,8,12)(14,19)(16,17)(20,27)(25,26);(1,2,3,4)(5,7,8,6,10,12,9,11)(13,15)(14,20)(16,26)(17,25)(18,24)(19,27)(21,28)(22,23)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=ehi,f^c=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239507
maximal normal subgroup of	#72239507
maximal quotient of	#72239511, #72239513

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17373);