#500271

order	512
nilpotency class	4
isoclinism	isoclinic class of #500268
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24058
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#61673	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

849 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 849 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407467);