#508231

order	512
nilpotency class	4
isoclinism	isoclinic class of #508024
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60685	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

739 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 739 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415427);