#512658

order	512
nilpotency class	5
isoclinism	isoclinic class of #512655
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	#124	C₂⁴
prime radical	#60820	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3499 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 818 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)(3,9,15,23)(4,8,17,21)(6,10,18,25)(7,12,19,26)(13,22,27,31)(14,24,28,32)(16,20,29,30);(2,8)(4,6)(5,18)(9,22)(10,12)(13,16)(15,27)(17,19)(20,24)(21,25)(23,32)(28,29);(1,3,7,14)(2,9,12,24)(4,16,6,13)(5,15,19,28)(8,20,10,22)(11,23,26,32)(17,29,18,27)(21,30,25,31);(1,4,7,6)(2,8,12,10)(3,13,14,16)(5,17,19,18)(9,22,24,20)(11,21,26,25)(15,27,28,29)(23,31,32,30)

Polycyclic presentation:

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

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, 419854);