#119553

order	512
nilpotency class	3
isoclinism	isoclinic class of #119550
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

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

normal subgroups only

This group has 781 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ei,g^b=gh,e^c=eh,g^c=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, 26749);