#184980

order	512
nilpotency class	3
isoclinism	isoclinic class of #184979
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3192 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 3192 conjugacy classes of subgroups — too many to draw. Only its 382 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=eh,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92176);