#500687

order	512
nilpotency class	4
isoclinism	isoclinic class of #500685
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24208
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

1202 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214466, #72214675, #72214714
maximal normal subgroup of	#72214466, #72214675, #72214714

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407883);