#148281

order	512
nilpotency class	4
isoclinism	isoclinic class of #13353
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2021
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#11087
prime radical	#3449	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

801 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282255
maximal normal subgroup of	#72282255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55477);