#500594

order	512
nilpotency class	4
isoclinism	isoclinic class of #500593
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24152
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

829 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214383, #72214645, #72214675
maximal normal subgroup of	#72214383, #72214645, #72214675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407790);