#512682

order	512
nilpotency class	5
isoclinism	isoclinic class of #512669
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	#60825	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3500 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

821 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 821 conjugacy classes of subgroups — too many to draw. Only its 184 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ehi,f^b=fgh,g^b=gi,d^c=di,e^c=ei,f^e=fhi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419878);