#119717

order	512
nilpotency class	3
isoclinism	isoclinic class of #119716
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

826 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 126 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)(3,14,16,26)(4,17,19,8)(5,13)(6,22)(7,12)(9,18,29,23)(10,21)(11,20)(15,28,31,25)(24,27)(30,32);(1,3,5,4)(2,8,13,26)(6,23,20,31)(7,16,21,19)(9,27,25,30)(10,14,12,17)(11,29,22,28)(15,24,18,32);(1,4,5,3)(2,9,10,28)(6,15,20,18)(7,19,21,16)(8,11,14,27)(12,29,13,25)(17,30,26,22)(23,24,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,e^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=ei,g^c=gh

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, 26913);