#512980

order	512
nilpotency class	5
isoclinism	isoclinic class of #512827
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33600	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

399 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,24)(13,27)(14,16)(17,20)(18,21)(19,25)(22,34)(23,28)(26,40)(29,36)(30,35)(31,37)(32,38)(33,39);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,15)(11,20)(12,21)(13,25)(16,28)(17,18)(19,34)(22,39)(23,37)(24,29)(26,38)(27,30)(31,32)(33,40)(35,36);(10,16)(15,28)(17,31)(18,32)(19,27)(24,40)(29,33)(30,34)

Polycyclic presentation:

2^9|c^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fg,g^a=gh,h^a=hi,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322434, #72322435, #72322440
maximal normal subgroup of	#72322434, #72322435, #72322440

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420176);