#110723

order	512
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,7,11,15)(4,6)(5,9)(8,12)(10,13,16,14)(18,23,25,28)(24,31,32,30);(2,6)(4,12)(7,13)(8,9)(10,16)(14,15)(17,18,21,23,22,25,20,28)(19,24,26,30,27,32,29,31);(1,3)(2,7,9,15)(4,10)(5,11)(6,13,8,14)(12,16)(17,19)(18,24)(20,29)(21,26)(22,27)(23,30)(25,32)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239604
maximal normal subgroup of	#72239604
maximal quotient of	#72239608, #72239871

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17919);