#119714

order	512
nilpotency class	3
isoclinism	isoclinic class of #119710
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 683 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26910);