#507792

order	512
nilpotency class	4
isoclinism	isoclinic class of #499903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3192
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#29409
prime radical	#13309	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

522 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,12)(4,9)(5,19)(6,11)(7,21)(8,14)(10,16)(13,28)(15,30)(17,26)(18,31)(20,27)(22,23)(24,29)(25,32);(3,13)(14,29)(15,22)(20,32);(1,3)(2,8)(4,14)(5,20)(6,13)(7,22)(9,23)(10,27)(11,24)(12,19)(15,16)(17,29)(18,32)(21,25)(26,30)(28,31);(1,4,16,5,6,17,7,18)(2,9,10,19,11,26,21,31)(3,14,15,20,13,29,22,32)(8,23,27,12,24,30,25,28)

Polycyclic presentation:

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

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, 414988);