#512875

order	512
nilpotency class	5
isoclinism	isoclinic class of #13677
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2196
commutator, Frattini, cosocle	#124	C₂⁴
socle	C₂	C₂×#2196
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1035 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1035 conjugacy classes of subgroups — too many to draw. Only its 112 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)(3,14)(4,16)(5,10)(6,22)(7,12)(8,27)(9,13)(11,21)(15,32)(17,24)(18,30)(19,31)(20,23)(25,28)(26,29);(1,3)(2,4)(5,15)(6,23)(7,25)(8,17)(9,18)(10,19)(11,30)(12,32)(13,20)(14,22)(16,21)(24,29)(26,31)(27,28);(1,4)(2,3)(5,19)(6,24)(7,14)(8,28)(9,20)(10,15)(11,31)(12,16)(13,18)(17,27)(21,32)(22,25)(23,29)(26,30);(1,5,9,8)(2,10,13,27)(3,15,18,17)(4,19,20,28)(6,7,21,26)(11,29,22,12)(14,32,30,24)(16,31,23,25)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420071);