#483990

order	512
nilpotency class	4
isoclinism	isoclinic class of #483987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=dhi,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285489, #72285695, #72286569, …
maximal normal subgroup of	#72285489, #72285695, #72286569, …
maximal quotient of	#72343467, #72343469, #72343498, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391186);