#507571

order	512
nilpotency class	4
isoclinism	isoclinic class of #499953
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3205
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#29254
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

501 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,14)(4,12)(5,8)(6,23)(7,13)(9,17)(10,21)(11,22)(15,28)(16,30)(18,29)(19,31)(20,26)(24,25)(27,32);(2,8,13,11)(4,20)(6,21)(9,27,28,25)(10,12,23,26)(14,30,31,29)(15,17)(24,32);(1,3)(2,9,13,28)(4,15)(5,16)(6,24)(7,19)(8,25,11,27)(10,14,23,31)(12,29,26,30)(17,20)(18,22)(21,32);(1,4,5,21,7,20,22,6)(2,10,11,12,13,23,8,26)(3,15,16,32,19,17,18,24)(9,14,27,29,28,31,25,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414767);