#148264

order	512
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2044
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

811 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279536
maximal normal subgroup of	#72279536
maximal quotient of	#72279561

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55460);