#147595

order	512
nilpotency class	4
isoclinism	isoclinic class of #147593
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10965
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#60367	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1692 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267973, #72267975, #72267977
maximal normal subgroup of	#72267973, #72267975, #72267977

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54791);