#508516

order	512
nilpotency class	4
isoclinism	isoclinic class of #508516
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29912
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

432 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415712);