#510080

order	512
nilpotency class	4
isoclinism	isoclinic class of #510079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31260
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24287	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 576 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)(3,6)(4,15)(5,7)(8,11)(9,12)(10,17)(13,20)(14,22)(16,32)(18,31)(19,28)(21,23)(24,26)(25,27)(29,30);(1,3,7,12,9,5,6,2)(4,16,17,27,20,31,19,30)(8,24,23,21,26,11,22,14)(10,18,15,25,28,32,13,29);(1,4)(2,10)(3,13)(5,15)(6,17)(7,19)(8,25)(9,20)(11,16)(12,28)(14,30)(18,23)(21,27)(22,32)(24,31)(26,29);(1,5,9,3)(2,7,12,6)(4,15,20,13)(8,21,26,14)(10,19,28,17)(11,23,24,22)(16,18,31,32)(25,27,29,30)

Polycyclic presentation:

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

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, 417276);