#510900

order	512
nilpotency class	4
isoclinism	isoclinic class of #510898
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32028
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28497	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 512 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)(3,13)(4,7)(5,12)(6,11)(8,10)(9,17)(14,26)(15,16)(18,29)(19,27)(20,24)(21,22)(23,32)(25,28)(30,31);(1,2,8,12,7,4,5,10)(3,14,17,29,19,32,16,31)(6,22,25,20,24,28,21,11)(9,26,13,30,15,23,27,18);(1,3,8,17,7,19,5,16)(2,9,10,27,4,15,12,13)(6,23,25,18,24,26,21,30)(11,29,28,14,20,31,22,32);(1,4,7,2)(3,15,19,9)(5,10,8,12)(6,20,24,11)(13,16,27,17)(14,18,32,30)(21,28,25,22)(23,31,26,29)

Polycyclic presentation:

2^9|b^2=ei,c^2=efh,d^2=i,e^2=i,f^2=i,g^2=hi,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, 418096);