#194973

order	512
nilpotency class	3
isoclinism	isoclinic class of #18162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238769, #72280469, #72280498
maximal normal subgroup of	#72238769, #72280469, #72280498

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102169);