#148007

order	512
nilpotency class	4
isoclinism	isoclinic class of #140675
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1964
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#11043
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,f^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^c=di,f^c=fi,g^c=gh,g^e=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280769, #72282225, #72282288
maximal normal subgroup of	#72280769, #72282225, #72282288

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55203);