#121344

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1444
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

510 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

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, 28540);