#495835

order	512
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₄×#459
prime radical	#33594	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

680 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=g,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,f^c=fh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225388, #72225389, #72225390, …
maximal normal subgroup of	#72225388, #72225389, #72225390, …
maximal quotient of	#72225507

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403031);