#148254

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

809 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281842, #72286452
maximal normal subgroup of	#72281842, #72286452

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55450);