#120631

order	512
nilpotency class	3
isoclinism	isoclinic class of #10518
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 416 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fi,g^b=gh,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27827);