#207006

order	512
nilpotency class	3
isoclinism	isoclinic class of #161570
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

886 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 886 conjugacy classes of subgroups — too many to draw. Only its 298 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,17)(12,21)(13,15)(14,19)(16,18)(20,23)(22,24)(25,26)(27,30,34,36)(28,32)(29,31)(33,37,39,40)(35,38);(2,5)(6,7)(9,11,18,24,13,20,12,19)(10,14,21,23,15,22,16,17)(25,27,29,34)(26,30)(28,33,35,39)(31,36)(32,37)(38,40);(1,3,4,8)(2,6,5,7)(10,15)(14,22)(16,21)(17,23)(26,31)(28,35)(30,36)(33,39);(25,28,29,35)(26,32,31,38)(27,33,34,39)(30,37,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fh,d^2=i,e^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114202);