#146590

order	512
nilpotency class	4
isoclinism	isoclinic class of #13096
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1898
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10735
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#585 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1629 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268000, #72280675, #72282724
maximal normal subgroup of	#72268000, #72280675, #72282724
maximal quotient of	#72267964

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53786);