#119559

order	512
nilpotency class	3
isoclinism	isoclinic class of #119556
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 757 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,g^b=gh,e^c=eh,g^c=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, 26755);