#496345

order	512
nilpotency class	4
isoclinism	isoclinic class of #496344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21794
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3508	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

758 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 188 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)(5,19,20,8)(6,11)(7,15)(10,13)(14,24)(16,21,32,28)(17,22)(18,25,29,26)(23,31,27,30);(1,3,13,14,6,17,4,15)(2,7,9,22,11,24,10,12)(5,16,29,30,20,32,18,31)(8,23,26,21,19,27,25,28);(2,8)(3,14)(4,13)(7,21)(9,25)(10,26)(11,19)(12,27)(15,17)(16,30)(18,29)(22,23)(24,28)(31,32);(2,9,11,10)(5,20)(7,22,24,12)(8,25,19,26)(16,32)(18,29)(21,23,28,27)(30,31)

Polycyclic presentation:

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

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, 403541);